Question

If $A=\left[\begin{array}{ccc}1& -2& 3\\ 2& 3& -1\\ -3& 1& 2\end{array}\right]B=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 2\\ 1& 2& 0\end{array}\right]$ Verify $A$ and $B$ commute w.r.t to matrix multiplication.

Answer 1

$A=\left[\begin{array}{ccc}1& -2& 3\\ 2& 3& -1\\ -3& 1& 2\end{array}\right],B=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 2\\ 1& 2& 0\end{array}\right]$

$A=\left[\begin{array}{ccc}1& -2& 3\\ 2& 3& -1\\ -3& 1& 2\end{array}\right],B=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 2\\ 1& 2& 0\end{array}\right]$

$=\left[\begin{array}{ccc}1+3& -2+3& 2-4\\ 2-1& 3-2& 4+6\\ -3+2& 1+4& -6+2\end{array}\right]$

$=\left[\begin{array}{ccc}4& 4& -2\\ 1& 1& 10\\ -1& 5& -4\end{array}\right]$

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

$=\left[\begin{array}{ccc}1-6& -2+2& 3+4\\ 2-6& 3+2& -1+4\\ 1+4& -2+6& 3-2\end{array}\right]$

$=\left[\begin{array}{ccc}-5& 0& 7\\ -4& 5& 3\\ 5& 4& 1\end{array}\right]$

$\therefore AB\ne BA$

$A$ & $B$ dont commute order matrix multiplication