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
$$A = P D P^{ - 1}$$
- $$A = \left[ \begin{array} { r r r r } 6 & 0 & 0 & 0 \\0 & 6 & 0 & 0 \\1 & - 4 & - 6 & 0 \\- 1 & 2 & 0 & - 6\end{array} \right]$$

A) $P = \left[ \begin{array} { r r r r } - 12 & - 24 & 0 & 0 \\ 6 & 6 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array} \right] , D = \left[ \begin{array} { r r r r } - 6 & 0 & 0 & 0 \\ 0 & - 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array} \right]$


B) $P = \left[ \begin{array} { r r r r } - 12 & 6 & 1 & 0 \\ - 24 & 6 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] , D = \left[ \begin{array} { r r r r } - 6 & 0 & 0 & 0 \\ 0 & - 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array} \right]$

C) Not diagonalizable

D)
$$P = \left[ \begin{array} { r r r r } - 12 & - 24 & 0 & 0 \\- 6 & - 6 & 0 & 0 \\1 & 0 & 1 & 0 \\0 & 1 & 0 & 1\end{array} \right] , D = \left[ \begin{array} { r r r r } 6 & 0 & 0 & 0 \\0 & 6 & 0 & 0 \\0 & 0 & - 6 & 0 \\0 & 0 & 0 & - 6\end{array} \right]$$