Question

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

Answer 1

$\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]$
$=\left[\begin{array}{ccc}1\times (-10+2\times 0+3\times 2& 1\times 1+2\times (-1)+3\times 3& 1\times 0+2\times 1+3\times 4\\ 0\times (-1)+1\times 0+0\times 2& 0\times 1+1(-1)+0\times 3& 0\times 0+1\times 1+0\times 4\\ 1\times (-1)+1\times 0+0\times 2& 1\times 1+1\times (-1)+0\times 3& 1\times 0+1\times 1+0\times 4\end{array}\right]$
$={\left[\begin{array}{ccc}5& 8& 14\\ 0& -1& 1\\ -1& 0& 1\end{array}\right]}_{3\times 3}$
$\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]$
$=\left[\begin{array}{ccc}-1\times 1+1\times 0+0\times 1& -1\times 2+1\times 1+0\times 1& -1\times 3+1\times 0+0\times 0\\ 0\times 1+-1\times 0+1\times 1& 0\times 2+1\times -1+1\times 1& 0\times 3+(-1)\times 0+1\times 0\\ 2\times 1+3\times 0+4\times 1& 2\times 2+3\times 1+4\times 1& 2\times 3+3\times 0+4\times 0\end{array}\right]$
$=\left[\begin{array}{ccc}-1& -1& -3\\ 1& 0& 0\\ 6& 11& 6\end{array}\right]$
hence $\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\ne \left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]$
thus verified.