If $A=\left[\begin{array}{ll}0 & 0 \\ 4 & 0\end{array}\right]$, find $A^{16}$
Given : $A=\left[\begin{array}{ll}0 & 0 \\ 4 & 0\end{array}\right]$
Here,
$A^{2}=A A$
$\Rightarrow A^{2}=\left[\begin{array}{ll}0 & 0 \\ 4 & 0\end{array}\right]\left[\begin{array}{ll}0 & 0 \\ 4 & 0\end{array}\right]$
$\Rightarrow A^{2}=\left[\begin{array}{ll}0+0 & 0+0 \\ 0+0 & 0+0\end{array}\right]$
$\Rightarrow A^{2}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
$A^{4}=A^{2} A^{2}$
$\Rightarrow A^{8}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
$\Rightarrow A^{8}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
$A^{16}=A^{8} A^{8}$
$\Rightarrow A^{16}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
$\therefore A^{16}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
Thus, $A^{16}$ is a null matrix.
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