5 It is given that $\lambda$ is an eigenvalue of the matrix $\mathbf { A }$ with $\mathbf { e }$ as a corresponding eigenvector, and $\mu$ is an eigenvalue of the matrix $\mathbf { B }$ for which $\mathbf { e }$ is also a corresponding eigenvector.\\
(i) Show that $\lambda + \mu$ is an eigenvalue of the matrix $\mathbf { A } + \mathbf { B }$ with $\mathbf { e }$ as a corresponding eigenvector.\\

The matrix $\mathbf { A }$, given by

$$\mathbf { A } = \left( \begin{array} { r r r } 
2 & 0 & 1 \\
- 1 & 2 & 3 \\
1 & 0 & 2
\end{array} \right)$$

has $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)$ and $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ as eigenvectors.\\
(ii) Find the corresponding eigenvalues.\\

The matrix $\mathbf { B }$ has eigenvalues 4, 5 and 1 with corresponding eigenvectors $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)$ and $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ respectively.\\
(iii) Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }$.\\

\hfill \mbox{\textit{CAIE FP1 2018 Q5}}