Question

If $A=\left(\begin{array}{cc}1& 3\\ 9& -6\end{array}\right)$, then verify $AI=IA=A$, where $I$ is the unit matrix of order $2$.

Answer 1

Given, $A=\left(\begin{array}{cc}1& 3\\ 9& -6\end{array}\right)$
$AI=\left(\begin{array}{cc}1& 3\\ 9& -6\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$
$=\left(\begin{array}{cc}1+0& 0+3\\ 9+0& 0-6\end{array}\right)=\left(\begin{array}{cc}1& 3\\ 9& -6\end{array}\right)=A$
$IA=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 3\\ 9& -6\end{array}\right)$
$=\left(\begin{array}{cc}1+0& 3-0\\ 0+9& 0-6\end{array}\right)=\left(\begin{array}{cc}1& 3\\ 9& -6\end{array}\right)=A$
Therefore, $AI=IA=A$