6 The matrix \(\mathbf { A }\) is given by
$$A = \left( \begin{array} { r r r }
2 & - 3 & - 7
0 & 5 & 7
0 & 0 & - 2
\end{array} \right) .$$
- Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
- Use the characteristic equation of \(\mathbf { A }\) to show that
$$\mathbf { A } ^ { 4 } = a \mathbf { A } ^ { 2 } + b \mathbf { I } ,$$
where \(a\) and \(b\) are integers to be determined.