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Question:

If $A=\left[\begin{array}{ll}0 & 0 \\ 4 & 0\end{array}\right]$, find $A^{16}$

Solution:

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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