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determinants of the block arrowhead matrices edyta hetmaniok micha l rozanski damian sl ota marcin szweda tomasz trawinski and roman witul a abstract the paper is devoted to the considerations ...

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            Determinants of the block arrowhead
            matrices
                                                   ´ ˙   ´
            Edyta HETMANIOK, Micha l ROZANSKI,
            Damian SL  OTA, Marcin SZWEDA,
                                ´
            Tomasz TRAWINSKI and Roman WITUL  A
            Abstract. The paper is devoted to the considerations on determinants of the block
            arrowhead matrices. At first we discuss, in the wide technical aspect, the motivation
            of undertaking these investigations. Next we present the main theorem concerning the
            formulasdescribing the determinants of the block arrowheadmatrices. We discuss also
            the application of these formulas by analyzing many specific examples. At the end of
            the paper we make an attempt to test the obtained formulas for the determinants of
            the block arrowhead matrices, but in case of replacing the standard inverses of the
            matrices by the Drazin inverses.
            Keywords: determinant, block matrix, arrowhead matrix, Drazin inverse.
            2010 Mathematics Subject Classification: 15A15, 15A23, 15A09.
            1. Technical introduction – motivation for discussion
              Observing the mathematical models, formulated for describing various dynami-
            cal systems used in the present engineering, one can notice that in many of them
            the block arrowhead matrices occur. One can find such matrices in the models of
            telecommunication systems [10], in robotics [18], in electrotechnics [17] and in auto-
            matic control [21]. In robotics, while modelling the dynamics of the kinematic chains
            of robots, or in electrotechnical problems, while modelling the electromechanical con-
            verters, there is often a need to formulate some equations in the coordinate systems
            different the ones in which the original model was formulated. Especially in the theory
            E. Hetmaniok, M. R´oz˙an´ski, D. S lota, M. Szweda, R. Witu la
            Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland,
            e-mail: {edyta.hetmaniok,michal.rozanski,damian.slota,marcin.szweda,roman.witula}@polsl.pl
            T. Trawin´ski
            Department of Mechatronics, Silesian University of Technology, Akademicka 10A, 44-100 Gliwice,
            Poland, e-mail: tomasz.trawinski@polsl.pl
            R. Witu la, B. Bajorska-Harapin´ska, E. Hetmaniok, D. S lota, T. Trawin´ski (eds.), Selected Problems
                                                       ´
            on Experimental Mathematics. Wydawnictwo Politechniki Sla¸skiej, Gliwice 2017, pp. 73–88.
                         74                               E. Hetmaniok, M. R´oz˙an´ski, D. S lota, M. Szweda, T. Trawin´ski and R. Witu la
                         of electromechanical converters one applies very widely the possibility of transform-
                         ing the mathematical models of the electromechanical converters form the natural
                         coordinate systems into the biaxial coordinate systems. The main goal of these trans-
                         formations is to eliminate the time dependence occurring in the coefficients of mutual
                         inductances, to increase the number of zero elements occurring in the matrices of
                         the appropriate mathematical model and, in consequence, to simplify this model and
                         to shorten the time needed for its solution. The forms of matrices transforming the
                         variables, occurring in the electromechanical converters, from the natural coordinate
                         systems into the biaxial coordinate systems can be deduced on the way of physical
                         reasoning. These matrices can be also found by using the methods of determining
                         the eigenvalues. Formulated matrices are applied, among others in the Park, Clark
                         and Stanley transformations [12, 9]. Elements of the transformation matrices contain
                         the eigenvalues of matrices of the electromechanical converter inductance coefficients.
                         These matrices can have, for example, the following forms [7]:
                                                                                   r cosϑ cosϑ cosϑ 
                                                                                        2               s1            s2            s3
                                                                     [K ] =                sinϑ             sinϑ          sinϑ         ;                                         (1)
                                                                          s             3               s1           s2            s3
                                                                                                    1             1             1
                                                                                                   √             √            √
                                                                                                      2            2             2
                                                                         1          cosα             cos2α            : : :    cos(Q −1)α                     
                                                                                              r                  r                         r             r
                                                                         0 −sinα −sin2α ::: −sin(Q −1)α 
                                                                                                r                  r                         r             r
                                                            r                                                                                                 
                                                                         1 cos2α                     cos4α            : : :   cos2(Q −1)α 
                                                                   2                           r                 r                           r            r
                                               [K ] =                                                                                                         ;                  (2)
                                                   r                     0 −sin2α −sin4α ::: −sin2(Q −1)α 
                                                                                                 r                 r                          r             r
                                                                 Qr  .                   .                 .           .                    .                 
                                                                         .               .                 .           .                    .                 
                                                                         .               .                 .           .                    .                 
                                                                             1            1                 1           1                    1
                                                                            √           √                 √            √                   √
                                                                              2            2                 2            2                   2
                         where
                                                                                         t
                                                              ϑ =ϑ +Z Ω dt;                                                                                                        (3)
                                                                s1          s10                x
                                                                                       0
                                                                                         t                                            t
                                                              ϑ =ϑ +Z Ω dt=ϑ +2π+Z Ω dt;                                                                                           (4)
                                                                s2          s20                x              s10         3                 x
                                                                                       0                                            0
                                                                                         t                                            t
                                                              ϑ =ϑ +Z Ω dt=ϑ +4π+Z Ω dt;                                                                                           (5)
                                                                s3          s30                x              s10         3                 x
                                                                                       0                                            0
                         whereasϑ                ; ϑ        ; ϑ        denotetheinitial angles between the axes of respective phases
                                            s10       s20        s30
                         of the stator and axis X of the biaxial system XY for moment of time t = 0; p means
                         the number of pole pairs, α denotes the rotor bar pitch and Ω describes the angular
                                                                            r                                                                 x
                         velocity of rotation of the biaxial system XY around the stator.
                               Matrix of the mutual and self inductance coefficients written in the natural coor-
                         dinate system, that is in the phase system, possesses the block structure – it is the
                         full matrix of the form
                                                                                        M=MssMsr:                                                                                (6)
                                                                                                     MT M
                                                                                                         sr       rr
                         Particular forms of matrices, of this type can be found, for example, in [7].
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Determinants of the block arrowhead matrices                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            75
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Method of transforming the matrix (6) of inductance coefficients into the new
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   coordinate system XY by using the transformation matrices (1) and (2) is as follows
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              XY                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    T
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Mss                                                                                                                                                                                                                                                                       =[Ks][Mss][Ks] ;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (7)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              XY                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         T
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Mrr                                                                                                                                                                                                                                                                       =[Kr][Mrr][Kr] ;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (8)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              XY                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      T
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Msr                                                                                                                                                                                                                                                                       =[Ks][Msr][Kr] :                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (9)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   By merging the obtained results we get the block arrowhead matrix [8], also in the
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   form presented in paper [17].
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Matrices (1) and (2) are often used for constructing the systems for regulating
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   the electromechanical converters, such as the various versions of the vector control
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   methods for the squirrel cage induction motors [13, 19]. Thus, there is a need for de-
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   veloping the effective methods of calculating the determinants of the block arrowhead
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   matrices. Only to this issue the second part of this paper (see [17]) will be devoted.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   2. The block arrowhead matrices
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Weuse the following notation in this paper:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       0 denotes the zero matrix of the respective size (of dimensions m×n),
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       1 denotes the identity matrix of the respective order (of order n).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Wepresent now the main result of this paper concerning the form of determinants
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   of the block arrowhead matrices. In Section 3 we examine few examples illustrating
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   the application of the obtained relations.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Theorem 2.1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      A                                                                                                                                                                                                                                                                              B                                                                                                                                                                                                   
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         1. Let us consider the block matrix M2 =                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            a×a                                                                                                                                                                                                                     a×b . The following implications
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   C                                                                                                                                                                                                                  D
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       hold:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  b×a                                                                                                                                                                                                                          b×b
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (a) If detD 6= 0, then detM2 = detDdet(A−BD−1C):
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (b) If detA 6= 0, then detM2 = detAdet(D −CA                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         B).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          In Remark 2.2, after the proof of the above theorem, we discuss the omitted
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          situation when detA = detD = 0 and a 6= b.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                A                                                                                                                                                                                                                                                                                         B                                                                                                                                                                                                                E                                                                                                                                                                                                   
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  a×a                                                                                                                                                                                                                     a×b                                                                                                                                                                                                            a×c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                C                                                                                                                                                                                                                                                                                          D                                                                                                                                                                                                                          0                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         2. Let us consider the block matrix M3 =                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   b×a                                                                                                                                                                                                                          b×b                                                                                                                                                                                                 b×c                                                                                                                                                                                               . The following impli-
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           F                                                                                                                                                                                                                           0                                                                                                                                                                                                     G
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       cations hold                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             c×a                                                                                                                                                                                                                  c×b                                                                                                                                                                                                                      c×c
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (a) If detA 6= 0 and det(D −CA                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   B)6= 0, then
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             −1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           detM3 =detAdet(D−CA B)det(G−FA E−
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           −1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      −1                                                                                                                                                                                          −1                                                                                                                                                                                                                       −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         −FA B(D−CA B) CA E): (10)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   76                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        E. Hetmaniok, M. R´oz˙an´ski, D. S lota, M. Szweda, T. Trawin´ski and R. Witu la
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Wenotethat if also detD 6= 0, then for the inverse of D−CA                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Bthefollowing
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Banachiewicz formula (see [1]), also called the Sherman-Morrison-Woodbury
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          (SMW)formula (for some more historical information see [22, subchapter 0.8]),
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          could be applied (see also S. Jose, K. C. Sivakumar, Moore-Penrose inverse of
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          perturbated operators on Hilbert Spaces, 119-131, in [2]):
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        −1                                                                                                                                                                                          −1                                                                                                                                                                                                                                                                              −1                                                                                                                                                                                                                                                                   −1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       −1                                                                                                                                                                                        −1                                                                                                                                                                                                                                  −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (D−CA B) =D +D C(A−BD C) BD :
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          The matrix A − BD−1C is of order a × a and the above formula is useful in
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          situations when a is much smaller than b and in all other situations when certain
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          structural properties of D are much more simple than of D −CA                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  B.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (b) If detD 6= 0 and detG 6= 0, then
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  −1                                                                                                                                                                                                                                                                                                                                                                                                 −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     detM3 =detDdetGdet(A−BD C−EG F):                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         (11)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (c) If either rankB+rankE ≤ b+c−1 or rankC D+rankF G ≤
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       D                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   G
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          b +c−1, then detM3 = 0.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       (d) If detG 6= 0, det(A−EG                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             F)6= 0 and D = 0, then
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    −1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   −1                                                                                                                                                                                         −1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    detM3 =detGdet(A−EG F)det(−C(A−EG F) B):                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  (12)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          In the sequel, if blocks B and C of M3 are the square matrices (that is a = b)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          and D = 0, then
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        a
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       detM3 =(−1) detGdetBdetC:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              (13)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          At last, if
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               −1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           −1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                rank A−EG F B =rank A−EG F +rank B
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          C                                                                                                                                                                                                                                                                                                   0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         C
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         −1                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      =rank C +rank A−EG F B 
						
									
										
									
																
													
					
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...Determinants of the block arrowhead matrices edyta hetmaniok micha l rozanski damian sl ota marcin szweda tomasz trawinski and roman witul a abstract paper is devoted to considerations on at rst we discuss in wide technical aspect motivation undertaking these investigations next present main theorem concerning formulasdescribing arrowheadmatrices also application formulas by analyzing many specic examples end make an attempt test obtained for but case replacing standard inverses drazin keywords determinant matrix inverse mathematics subject classication introduction discussion observing mathematical models formulated describing various dynami cal systems used engineering one can notice that them occur nd such telecommunication robotics electrotechnics auto matic control while modelling dynamics kinematic chains robots or electrotechnical problems electromechanical con verters there often need formulate some equations coordinate dierent ones which original model was especially theory e ...

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