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# If $A=\begin{bmatrix}2&0&0\\0&2&0\\0&0&2\end{bmatrix}$ then $A^5=$

$\begin{array}{1 1}(A)\;5A&(B)\;10A\\(C)\;16A&(D)\;32A\end{array}$

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A)
$A=\begin{bmatrix}2&0&0\\0&2&0\\0&0&2\end{bmatrix}$
This can be written as $2\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$
$\Rightarrow 2I$
$A=2I$
$A^2=2I\times 2I$
$\Rightarrow 4I^2$
$\Rightarrow 4I$
$A^4=4I\times 4I$
$\Rightarrow 16I^2$
$\Rightarrow 16I$
$A^5=16I\times A$
$\Rightarrow 16A$
Hence (C) is the correct answer.