If P = begin{bmatrix} 2 & -2 & -4 -1 & 3 & 4 | vedclass

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(A) Given,$P = \begin{bmatrix} 2 & -2 & -4 \ -1 & 3 & 4 \ 1 & -2 & -3 \end{bmatrix}$.First,we calculate $P^2 = P \cdot P$:$P^2 = \begin{bmatrix} 2 &

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(A) Given,$P = \begin{bmatrix} 2 & -2 & -4 \ -1 & 3 & 4 \ 1 & -2 & -3 \end{bmatrix}$.First,we calculate $P^2 = P \cdot P$:$P^2 = \begin{bmatrix} 2 & -2 & -4 \ -1 & 3 & 4 \ 1 & -2 & -3 \end{bmatrix} \begin{bmatrix} 2 & -2 & -4 \ -1 & 3 & 4 \ 1 & -2 & -3 \end{bmatrix}$$P^2 = \begin{bmatrix} (4+2-4) & (-4-6+8) & (-8-8+12) \ (-2-3+4) & (2+9-8) & (4+12-12) \ (2+2-3) & (-2-6+6) & (-4-8+9) \end{bmatrix}$$P^2 = \begin{bmatrix} 2 & -2 & -4 \ -1 & 3 & 4 \ 1 & -2 & -3 \end{bmatrix} = P$.Since $P^2 = P$,it follows that $P^3 = P^2 \cdot P = P \cdot P = P^2 = P$.By induction,$P^n = P$ for all positive integers $n \ge 1$.Therefore,$P^5 = P$.

If $P = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$,then $P^5$ is equal to

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