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.

\begin{enumerate}[label=(\roman*)]
\item Show that $\lambda + \mu$ is an eigenvalue of the matrix $\mathbf{A} + \mathbf{B}$ with $\mathbf{e}$ as a corresponding eigenvector. [2]
\end{enumerate}

The matrix $\mathbf{A}$, given by
$$\mathbf{A} = \begin{pmatrix} 2 & 0 & 1 \\ -1 & 2 & 3 \\ 1 & 0 & 2 \end{pmatrix}$$

has $\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$, $\begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ as eigenvectors.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the corresponding eigenvalues. [3]
\end{enumerate}

The matrix $\mathbf{B}$ has eigenvalues $4$, $5$ and $1$ with corresponding eigenvectors $\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$, $\begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ respectively.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $(\mathbf{A} + \mathbf{B})^3 = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE FP1 2018 Q5 [8]}}