Diagonalize the Matrix A, If Possible $$A = P D P^{ - 1}$$

Multiple Choice

Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that
$$A = P D P^{ - 1}$$
- $A=\left[\begin{array}{rrrr}9 & 0 & 0 & 0 \\0 & 9 & 0 & 0 \\-16 & 4 & 1 & 16 \\0 & 0 & 0 & 9\end{array}\right]$

A) $P=\left[\begin{array}{rrrr}2 & 0 & -2 & 1 \\0 & 2 & 1 & 0 \\1 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{array}\right], D=\left[\begin{array}{llll}9 & 0 & 0 & 0 \\0 & 9 & 0 & 0 \\0 & 0 & 9 & 0 \\0 & 0 & 0 & 1\end{array}\right]$

B) Not diagonalizable

C) $P=\left[\begin{array}{rrrr}4 & -2 & 1 & 0 \\8 & -2 & 0 & 0 \\1 & 0 & 1 & 1 \\0 & 1 & 1 & 0\end{array}\right], D=\left[\begin{array}{llll}9 & 0 & 0 & 0 \\0 & 9 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{array}\right]$

D) $P=\left[\begin{array}{rrrr}2 & 0 & 1 & 0 \\0 & 2 & 0 & 0 \\-2 & 1 & 0 & 1 \\1 & 0 & 1 & 0\end{array}\right], D=\left[\begin{array}{llll}9 & 0 & 0 & 0 \\0 & 9 & 0 & 0 \\0 & 0 & 9 & 0 \\0 & 0 & 0 & 1\end{array}\right]$