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Question

# (a) verify that A is diagonalizable by finding $P^{ - 1}AP,$ and (b) use the result of part (a) and Theorem to find the eigenvalues of A.$A=\left[\begin{array}{rrr} {-1} & {1} & {0} \\ {0} & {3} & {0} \\ {4} & {-2} & {5} \end{array}\right], P=\left[\begin{array}{rrr} {0} & {1} & {-3} \\ {0} & {4} & {0} \\ {1} & {2} & {2} \end{array}\right]$

Solution

Verified
Step 1
1 of 4

Begin by determining $P^{-1}$. Since $PP^{-1}=I$,

$\begin{bmatrix} 0&1&-3\\ 0&4&0\\ 1&2&2 \end{bmatrix} \begin{bmatrix} x_{11}&x_{12}&x_{13}\\ x_{21}&x_{22}&x_{23}\\ x_{31}&x_{32}&x_{33} \end{bmatrix}= \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}$

from which

$\begin{cases} x_{21}-3x_{31}=1\\ x_{22}-3x_{32}=0\\ x_{23}-3x_{33}=0\\ 4x_{21}=0\\ 4x_{22}=1\\ 4x_{23}=0\\ x_{11}+2x_{21}+2x_{31}=0\\ x_{12}+2x_{22}+2x_{32}=0\\ x_{13}+2x_{23}+2x_{33}=1 \end{cases}$

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