9 It is given that $\mathbf { e }$ is an eigenvector of the matrix $\mathbf { A }$, with corresponding eigenvalue $\lambda$.\\
(i) 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.\\
(ii) Find, in terms of $n$, a non-singular matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q9}}