View Policies Current Attempt in Progress Let $A = \begin{bmatrix} 1 & -2 & 8 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ and $P = \begin{bmatrix} 1 & 1 & -4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Given that P diagonalizes A, then compute each of the following powers of A. (a) $A^{1140} = \begin{bmatrix} ? & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{bmatrix}$ (b) $A^{-1140} = \begin{bmatrix} ? & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{bmatrix}$
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Since P diagonalizes A, the columns of P are the eigenvectors of A, and the diagonal entries of P are the corresponding eigenvalues. Show more…
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Let $$A=\left[\begin{array}{rrr}1 & -2 & 8 \\0 & -1 & 0 \\0 & 0 & -1\end{array}\right] \quad \text { and } \quad P=\left[\begin{array}{rrr}1 & -4 & 1 \\1 & 0 & 0 \\0 & 1 & 0\end{array}\right]$$ Confirm that $P$ diagonalizes $A,$ and then compute each of the following powers of $A .$ (a) $A^{1000}$ (b) $A^{-1000}$ (c) $A^{2301}$ (d) $A^{-2301}$
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