Question

If $A=\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]$. Check commutativity of the two matrices

Answer 1

$A=\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right],B=\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]AB=\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]=\left[\begin{array}{cc}2-8+6& 3-10+3\\ -8+8+10& -12+10+5\end{array}\right]=\left[\begin{array}{cc}0& -4\\ 10& 3\end{array}\right]BA=\left[\begin{array}{cc}2& 3\\ 4& 5\\ 2& 1\end{array}\right]\left[\begin{array}{ccc}1& -2& 3\\ -4& 2& 5\end{array}\right]=\left|\begin{array}{ccc}2-12& -4+6& 6+15\\ 4-20& -8+10& 12+25\\ 2-4& -4+2& -6+5\end{array}\right|\left[\begin{array}{ccc}-10& 2& 21\\ -16& 2& 37\\ -2& -2& -1\end{array}\right]AB\ne BA$