Question

If $A=\left[\begin{array}{c}2\\ 0\\ 0\end{array}\begin{array}{c}0\\ 2\\ 0\end{array}\begin{array}{c}0\\ 0\\ 2\end{array}\right]$ then $A^5=$

• A. $5A$
• B. $10A$
• C. $16A$
• D. $32A$

Answer 1

Answer: (C)

$16A$

Given, $A=\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$
$\Rightarrow A=2\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow A=2I$
$\Rightarrow A^5=2^5{I}^5$
$\Rightarrow A^5=2^{5}I$
$\Rightarrow A^5=2^4.2I$
$\Rightarrow A^5=2^4.2\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow A^5=16\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$.
$\Rightarrow A^5=16A$
Option $C$ is correct.