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Question
If \[A = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\] , find A4.
Solution
\[Here, \]
`A^2 `= AA
\[ \Rightarrow A^2 = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}9 + 0 & 0 + 0 \\ 0 + 0 & 0 + 9\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}9 & 0 \\ 0 & 9\end{bmatrix}\]
\[Now, \]
\[ A^4 = A^2 A^2 \]
\[ \Rightarrow A^4 = \begin{bmatrix}9 & 0 \\ 0 & 9\end{bmatrix}\begin{bmatrix}9 & 0 \\ 0 & 9\end{bmatrix}\]
\[ \Rightarrow A^4 = \begin{bmatrix}81 + 0 & 0 + 0 \\ 0 + 0 & 0 + 81\end{bmatrix}\]
\[ \Rightarrow A^4 = \begin{bmatrix}81 & 0 \\ 0 & 81\end{bmatrix}\]
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