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  4. Let A=[a&0&0 0&a&0 0&0&a], then A n is equal to

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Q. Let $A=\begin{bmatrix}a&0&0\\ 0&a&0\\ 0&0&a\end{bmatrix}$, then $A ^{ n }$ is equal to

Matrices

Solution:

$A^2=\begin{bmatrix}a&0&0\\ 0&a&0\\ 0&0&a\end{bmatrix} \begin{bmatrix}a&0&0\\ 0&a&0\\ 0&0&a\end{bmatrix}=\begin{bmatrix}a^2&0&0\\ 0&a^2&0\\ 0&0&a^2\end{bmatrix}$
$A^3=\begin{bmatrix}a^2&0&0\\ 0&a^2&0\\ 0&0&a^2\end{bmatrix}\begin{bmatrix}a&0&0\\ 0&a&0\\ 0&0&a\end{bmatrix}=\begin{bmatrix}a^3&0&0\\ 0&a^3&0\\ 0&0&a^3\end{bmatrix}$