5 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
18 & 5 & - 11
8 & 6 & - 4
32 & 10 & - 20
\end{array} \right)$$
- Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 4 \lambda ^ { 2 } - 20 \lambda + 48 = 0\) and hence find the eigenvalues of \(\mathbf { A }\).
- Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).