3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 3 & 0
3 & - 2 & - 1
0 & - 1 & 1 \end{array} \right)\).
- Show that the characteristic equation of \(\mathbf { M }\) is
$$\lambda ^ { 3 } - 13 \lambda + 12 = 0 .$$
- Find the eigenvalues and corresponding eigenvectors of \(\mathbf { M }\).
- Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that
$$\mathbf { M } ^ { n } = \mathbf { P D P } ^ { - 1 } .$$
(You are not required to calculate \(\mathbf { P } ^ { - 1 }\).)