9 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 \(\mathbf { A } ^ { 2 }\), with corresponding eigenvalue \(\lambda ^ { 2 }\).
The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by
$$\mathbf { A } = \left( \begin{array} { c c c }
n & 1 & 3
0 & 2 n & 0
0 & 0 & 3 n
\end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + n \mathbf { I } ) ^ { 2 }$$
where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix and \(n\) is a non-zero integer. - Find, in terms of \(n\), a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).