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Verify that \( ( AB ) C = A ( BC ) \) if \( A=\left[\begin{array}{ll}1 & 2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] \) and \( C=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 0\end{array}\right] \)

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AB = [1  2 ]    [2  1 ]   =  [2+6      1+4]   =   [8     5]

         [3  1 ]    [3  2 ]       [6+3       3+2]        [9     5]

(AB)C  =  [8    5]   [1   2    3] =[8+15  16+5  24+0]

                [9     5]   [3   1   0]    [9+15  18+5  27+0]

                                               = [23     21     24]. 

                                                  [24     23     27]. - I

BC =  [2    1] [1    2   3] = [2+3    4+1   6+0]

          [3     2] [3   1   0].    [3+6.   6+2.  9+0]

                                        = [5.     5.     6]

                                           [9.     8.      9]

A(BC) = [1   2].  [5.   5.   6] = [ 5+18  5+16  6+18]

              [3    1].  [9.   8.   9]    [15+9  15+8  18+9]

                                              = [23.    21.   24]

                                                [24.     23.   27]__II

From I and II (AB) C = A (BC)


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