Show that two matrices can be diagonalized

Prove that the matrices A = \begin{bmatrix} 5 & 6 & 0 \\ -3 & -4 & 0 \\ -2 & 0 & 1 \end{bmatrix} and B = \begin{bmatrix} 3 & -1 & 2 \\ -10 & 6 & -14 \\ -6 & 3 & -7 \end{bmatrix} are similar. Show that both can be diagonalized and give matrices P and Q such that P^{-1}AP and Q^{-1}BQ are diagonal.


Let P = \begin{bmatrix} -1 & -2 & 0 \\ 5 & 6 & 1 \\ -1 & -1 & 0 \end{bmatrix} and Q = \begin{bmatrix} 2 & -1 & 3 \\ -1 & 1 & -2 \\ -2 & 1 & -2 \end{bmatrix}. Evidently we have P^{-1}AP = Q^{-1}BQ = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}. In particular, A and B are similar and both are diagonalizable.

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