$(25$ pts$)$ Diagonalize the following matrix $A.$ That is$,$ find an invertible matrix

$P$ and diagonal matrix $D$ such that $A=PD{P}^{-1}.$

$A=\left[\begin{array}{ccc}4& 1& -2\\ 2& 5& 4\\ 0& 0& 5\end{array}\right].$

Justify your answer.

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