Question

Show that $\left(ii\right)\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\ne \left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]$

Answer 1

Here,$\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]=\left[\begin{array}{ccc}-1+0+6& 1-2+9& 0+2+12\\ 0+0+0& 0+(-1)+0& 0+1+0\\ -1+0+0& 1-1+0& 0+1+0\end{array}\right]=\left[\begin{array}{ccc}5& 8& 14\\ 0& -1& 1\\ -1& 0& 1\end{array}\right]and\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]$
$=\left[\begin{array}{ccc}-1+0+0& -2+1+0& -3+0+0\\ 0+0+1& 0-1+1& 0+0+0\\ 2+0+4& 4+3+4& 6+0+0\end{array}\right]=\left[\begin{array}{ccc}-1& -1& -3\\ 1& 0& 0\\ 6& 11& 6\end{array}\right]$
Hence, the required result is verified.