Diagonalize the matrix \( A \), if possible. That is, find an invertible matrix \( P \) and a diagonal matrix \( D \) such that \( A=P D P^{-1} \). \[ A=\left[\begin{array}{rrr} 1 & 1 & 4 \\ 0 & -4 & 0 \\ -5 & -1 & -8 \end{array}\right] \] A. Not diagonalizable \( P=\left[\begin{array}{rrr}1 & 0 & -1 \\ 0 & -4 & 0 \\ 1 & 1 & 1\end{array}\right], D=\left[\begin{array}{rrr}-4 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -3\end{array}\right] \) \( P=\left[\begin{array}{rrr}-4 & -1 & -1 \\ 0 & 5 & 0 \\ 5 & 0 & 1\end{array}\right], D=\left[\begin{array}{rrr}-4 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & -3\end{array}\right] \) \( P=\left[\begin{array}{rrr}1 & -9 & -1 \\ -9 & -4 & 0 \\ 1 & -4 & 1\end{array}\right], D=\left[\begin{array}{rrr}-4 & 1 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & -3\end{array}\right] \)
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The eigenvalues are the roots of the characteristic polynomial, which is found by subtracting lambda from the diagonal elements of A and then taking the determinant. The characteristic polynomial of A is given by: det(A - λI) = det([[1-λ, 1, 4], [0, -4-λ, 0], Show more…
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