Question

If $A=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right]$, Show that $AB\ne BA$.

Answer 1

$A=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right]$ and $\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right]$

$AB=\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right]\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right]$

$AB=\left[\begin{array}{cc}4+4& 12+2\\ 3+8& 9+4\end{array}\right]=\left[\begin{array}{cc}8& 14\\ 11& 13\end{array}\right]$

$BA=\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right]\left[\begin{array}{cc}4& 1\\ 3& 2\end{array}\right]$

$BA=\left[\begin{array}{cc}4+9& 1+6\\ 16+6& 4+4\end{array}\right]=\left[\begin{array}{cc}13& 7\\ 22& 8\end{array}\right]$

Hence, $AB\ne BA$.