5 It is given that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\) with corresponding eigenvalue \(\lambda\).
- Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 3 }\) and state the corresponding eigenvalue.
It is given that
$$\mathbf { A } = \left( \begin{array} { r r }
2 & 0
- 1 & 3
\end{array} \right) .$$ - Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that
$$\mathbf { A } ^ { 3 } + \mathbf { I } = \mathbf { P } \mathbf { D } \mathbf { P } ^ { - 1 }$$
where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.