If A = begin{bmatrix} 2 & 0 & 0 0 & -2 & 0 | vedclass

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(A) We know that $A^4 A^{-1} = A^{4-1} = A^3$. Since $A$ is a diagonal matrix,$A^n = \begin{bmatrix} a_{11}^n & 0 & 0 \ 0 & a_{22}^n & 0 \ 0 & 0 & a

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$\begin{bmatrix} 8 & 0 & 0 \ 0 & -8 & 0 \ 0 & 0 & -1 \end{bmatrix}$

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(A) We know that $A^4 A^{-1} = A^{4-1} = A^3$. Since $A$ is a diagonal matrix,$A^n = \begin{bmatrix} a_{11}^n & 0 & 0 \ 0 & a_{22}^n & 0 \ 0 & 0 & a_{33}^n \end{bmatrix}$. Given $A = \begin{bmatrix} 2 & 0 & 0 \ 0 & -2 & 0 \ 0 & 0 & -1 \end{bmatrix}$,we calculate $A^3$: $A^3 = \begin{bmatrix} 2^3 & 0 & 0 \ 0 & (-2)^3 & 0 \ 0 & 0 & (-1)^3 \end{bmatrix} = \begin{bmatrix} 8 & 0 & 0 \ 0 & -8 & 0 \ 0 & 0 & -1 \end{bmatrix}$. Thus,$A^4 A^{-1} = \begin{bmatrix} 8 & 0 & 0 \ 0 & -8 & 0 \ 0 & 0 & -1 \end{bmatrix}$.

If $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{bmatrix}$,then $A^4 A^{-1} = $

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