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Question
If \[A = \begin{bmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{bmatrix},\text{ then }A^5 =\] ____________ .
Options
5A
10A
16A
32A
Solution
16A
\[A = \begin{bmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{bmatrix}\]
\[ \Rightarrow A = 2\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]
\[ \Rightarrow A = 2I\]
\[ \Rightarrow A^5 = \left( 2I \right)^5 \]
\[ \Rightarrow A^5 = 16 \times 2I\]
\[ \Rightarrow A^5 = 16\begin{bmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{bmatrix}\]
\[ \Rightarrow A^5 = 16A\]
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