If A= -3&0 0&-3 , find A^4. - Vidyadip

Question

If $\text{A}=\begin{bmatrix}-3&0\\0&-3\end{bmatrix}$, find $A^4$.

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Answer

Here,
$\text{A}^2 = \text{AA}$
$\Rightarrow\text{A}^2=\begin{bmatrix}-3&0\\0&-3\end{bmatrix}\begin{bmatrix}-3&0\\0&-3\end{bmatrix}$
$\Rightarrow\text{A}^2=\begin{bmatrix}9+0&0+0\\0+0&0+9\end{bmatrix}$
$\Rightarrow\text{A}^2=\begin{bmatrix}9&0\\0&9\end{bmatrix}$
Now,
$\text{A}^4 = \text{A}^2\cdot\text{A}^2$
$\Rightarrow\text{A}^4=\begin{bmatrix}9&0\\0&9\end{bmatrix}\begin{bmatrix}9&0\\0&9\end{bmatrix}$
$\Rightarrow\text{A}^4=\begin{bmatrix}81+0&0+0\\0+0&0+81\end{bmatrix}$
$\Rightarrow\text{A}^4=\begin{bmatrix}81&0\\0&81\end{bmatrix}$

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