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
\(\mathbf{A}=\left(\begin{array}{rrr}
2 & 0 & 1 \\
-1 & 2 & 3 \\
1 & 0 & 2
\end{array}\right)\)
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 \((\mathbf{A}+\mathbf{B})^{3}=\mathbf{P D P}^{-1}\).