Question
If $A =\left[\begin{array}{cc}1 & 2 \\ -2 & 3\end{array}\right], B =\left[\begin{array}{ll}2 & 1 \\ 2 & 3\end{array}\right] C =\left[\begin{array}{cc}-3 & 1 \\ 2 & 0\end{array}\right]$ verify that $(AB)C = A(BC),$
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Answer
$A B=\left[\begin{array}{cc}1 & 2 \\ -2 & 3\end{array}\right]\left[\begin{array}{ll}2 & 1 \\ 2 & 3\end{array}\right] $
$ =\left[\begin{array}{cc}2+4 & 1+6 \\ -4+6 & -2+9\end{array}\right]=\left[\begin{array}{ll}6 & 7 \\ 2 & 7\end{array}\right]$
$( AB ) C =\left[\begin{array}{ll}6 & 7 \\ 2 & 7\end{array}\right]\left[\begin{array}{cc}-3 & 1 \\ 2 & 0\end{array}\right] $
$=\left[\begin{array}{cc}-18+14 & 6+0 \\ -6+14 & 2+0\end{array}\right]=\left[\begin{array}{cc}-4 & 6 \\ 8 & 2\end{array}\right] $
Now $, BC =\left[\begin{array}{ll}2 & 1 \\ 2 & 3\end{array}\right]\left[\begin{array}{cc}-3 & 1 \\ 2 & 0\end{array}\right] $
$ =\left[\begin{array}{cc}-6+2 & 2+0 \\ -6+6 & 2+0\end{array}\right]=\left[\begin{array}{cc}-4 & 2 \\ 0 & 2\end{array}\right]$
$ A(B C)=\left[\begin{array}{cc} 1 & 2 \\ -2 & 3 \end{array}\right]\left[\begin{array}{cc} -4 & 2 \\
0 & 2 \end{array}\right] $
$=\left[\begin{array}{cc} -4+0 & 2+4 \\ 8+0 & -4+6 \end{array}\right]=\left[\begin{array}{cc} -4 & 6 \\ 8 & 2
\end{array}\right]$
Hence, $(A B) C=A(B C)$.
$ =\left[\begin{array}{cc}2+4 & 1+6 \\ -4+6 & -2+9\end{array}\right]=\left[\begin{array}{ll}6 & 7 \\ 2 & 7\end{array}\right]$
$( AB ) C =\left[\begin{array}{ll}6 & 7 \\ 2 & 7\end{array}\right]\left[\begin{array}{cc}-3 & 1 \\ 2 & 0\end{array}\right] $
$=\left[\begin{array}{cc}-18+14 & 6+0 \\ -6+14 & 2+0\end{array}\right]=\left[\begin{array}{cc}-4 & 6 \\ 8 & 2\end{array}\right] $
Now $, BC =\left[\begin{array}{ll}2 & 1 \\ 2 & 3\end{array}\right]\left[\begin{array}{cc}-3 & 1 \\ 2 & 0\end{array}\right] $
$ =\left[\begin{array}{cc}-6+2 & 2+0 \\ -6+6 & 2+0\end{array}\right]=\left[\begin{array}{cc}-4 & 2 \\ 0 & 2\end{array}\right]$
$ A(B C)=\left[\begin{array}{cc} 1 & 2 \\ -2 & 3 \end{array}\right]\left[\begin{array}{cc} -4 & 2 \\
0 & 2 \end{array}\right] $
$=\left[\begin{array}{cc} -4+0 & 2+4 \\ 8+0 & -4+6 \end{array}\right]=\left[\begin{array}{cc} -4 & 6 \\ 8 & 2
\end{array}\right]$
Hence, $(A B) C=A(B C)$.
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