Some Viewpoints on Matrix Multiplication.
                                          
                         ab             ef
                Let A =        and B =        .Then
                         cd             gh
                                                                
                               AB= ab ef= ae+bg af+bh
                                      cdgh ce+dg cf+dh
            There are many ways to interpret this remarkable operation. Perhaps the most natural
            one being the interpretation of matrix multiplication as function composition (of linear
            functions) from which the associativity follows directly. But the following formulas suggest
            other interpretations.
                                     a      b    a    b
                                  =e c +g d        f  c  +h d 
            i.e. multiplication of A on the right by a matrix corresponds to column operations on A.
                                                           
                                           a[ef]+b[gh]
                                        =                  
                                                           
                                           c[ef]+d[gh]
            i.e. multiplication of B on the left by a matrix corresponds to row operations on B.
                           =ae af+bg bh=a[ef]+b[gh]
                               ce  cf     dg  dh     c           d
            i.e. we can express AB as a sum of matrices each of rank 1.
                In Matrix Algebra, you can choose which interpretation you wish to use at a given
            point in an argument.
                What follows are the same calculations for 3 ×3 matrices.
                                                 
                         abc                 qrs
                                                 
                Let A =  defandB= tuv.Then
                         ghi                xyz
                                                                               
                       abc qrs aq+bt+cx ar+bu+cy as+bv+cz
                                                                               
                AB= def tuv= dq+et+fx dr+eu+fy ds+ev+fz
                       ghi xyz gq+ht+ix gr+hu+iy gs+hv+iz
                   a      b     c    a     b      c    a     b     c
                                                                                        
            AB=qd+te+xf rd+ue+yf sd+ve+zf
                     g       h        i      g       h       i       g       h       i  
                                                  1
              i.e. multiplication of A on the right by a matrix corresponds to column operations on A.
                                                                              
                                        a[qrs]+b[tuv]+c[xyz]
                                                                              
                                                                              
                                AB= d[qrs]+e[tuv]+f[xyz]
                                                                              
                                                                              
                                        g[qrs]+h[tuv]+i[xyz]
              i.e. multiplication of B on the left by a matrix corresponds to row operations on B.
                                  aq ar as bt bu bv cx cy cz
                            AB=dq dr ds+et eu ev+fx fy fz
                                    gq gr gs         ht  hu hv         ix   iy  iz
                                a               b              c
                                                                
                              = d [qrs]+ e [tuv]+ f [xyz]
                                  g                h                 i
              i.e. we can express AB as a sum of matrices each of rank 1.
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