If $A=\left[\begin{array}{rr}-3 & 0 \\ 0 & -3\end{array}\right]$, find $A^{4}$
Here,
$A^{2}=A A$
$\Rightarrow A^{2}=\left[\begin{array}{cc}-3 & 0 \\ 0 & -3\end{array}\right]\left[\begin{array}{cc}-3 & 0 \\ 0 & -3\end{array}\right]$
$\Rightarrow A^{2}=\left[\begin{array}{ll}9+0 & 0+0 \\ 0+0 & 0+9\end{array}\right]$
$\Rightarrow A^{2}=\left[\begin{array}{ll}9 & 0 \\ 0 & 9\end{array}\right]$
Now,
$A^{4}=A^{2} A^{2}$
$\Rightarrow A^{4}=\left[\begin{array}{ll}9 & 0 \\ 0 & 9\end{array}\right]\left[\begin{array}{ll}9 & 0 \\ 0 & 9\end{array}\right]$
$\Rightarrow A^{4}=\left[\begin{array}{cc}81+0 & 0+0 \\ 0+0 & 0+81\end{array}\right]$
$\Rightarrow A^{4}=\left[\begin{array}{cc}81 & 0 \\ 0 & 81\end{array}\right]$
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