6 The matrix \(\mathbf { P }\) is given by
$$\mathbf { P } = \left( \begin{array} { r r r }
1 & 6 & 6
0 & 2 & 6
0 & 0 & - 3
\end{array} \right) .$$
- Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
- Find the matrix \(\mathbf { A }\) such that
$$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { l l l }
4 & 0 & 0
0 & 5 & 0
0 & 0 & 6
\end{array} \right) .$$ - State the eigenvalues and corresponding eigenvectors of \(\mathbf { A } ^ { 3 }\).