Let
\begin{equation*}
A = \begin{bmatrix} 1 & -48 & -48 \\ -4 & -43 & -48 \\ 4 & 48 & 53 \end{bmatrix}.
\end{equation*}
The characteristic polynomial of A is $(1 - \lambda)(5 - \lambda)^2$.
(a) Determine a basis for the eigenspace $E_5$.
(b) A basis for the eigenspace $E_1$ is \{ $(-1, -1, 1)$ \}. Find an invertible matrix $P$ and a diagonal matrix $D$ such that $D = P^{-1}AP$, if they exist, or explain why they do not exist.
