The matrix \(\mathbf{P}\) is given by
\[\mathbf{P}=\left(\begin{array}{rrr}
1 & 6 & 6 \\
0 & 2 & 6 \\
0 & 0 & -3
\end{array}\right) .\]
(a) Use the characteristic equation of \(\mathbf{P}\) to find \(\mathbf{P}^{-1}\).

(b) Find the matrix \(\mathbf{A}\) such that
\[\mathbf{P}^{-1} \mathbf{A} \mathbf{P}=\left(\begin{array}{ccc}
4 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 6
\end{array}\right) .\]
(c) State the eigenvalues and corresponding eigenvectors of \(\mathbf{A}^{3}\).