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Question 5.2.1: Let A = [ 1 2 0 2 1 0 0 0 -3] Show that A is diagonalizable ...

Let
A = \begin{bmatrix} 1&2&0 \\2&1&0 \\ 0&0&-3 \end{bmatrix}

Show that A is diagonalizable with
P = \begin{bmatrix} 1&1&0 \\-1&1&0 \\ 0&0&1 \end{bmatrix}

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The inverse matrix is given by

P^{-1}= \begin{bmatrix} \frac{1}{2} &-\frac{1}{2} &0 \\ \frac{1}{2} &\frac{1}{2} &0 \\ 0&0&1 \end{bmatrix}

so that

P^{-1}AP=\begin{bmatrix} -1&0&0 \\0&3&0 \\ 0&0&-3 \end{bmatrix}

Therefore, the matrix A is diagonalizable

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