The vector \(\mathbf { e }\) is an eigenvector of the square matrix \(\mathbf { G }\). Show that
- \(\mathbf { e }\) is an eigenvector of \(\mathbf { G } + k \mathbf { I }\), where \(k\) is a scalar and \(\mathbf { I }\) is an identity matrix,
- \(\mathbf { e }\) is an cigenvector of \(\mathbf { G } ^ { 2 }\).
Find the eigenvalues, and corresponding eigenvectors, of the matrices \(\mathbf { A }\) and \(\mathbf { B } ^ { 2 }\), where
$$\mathbf { A } = \left( \begin{array} { r r r }
3 & - 3 & 0
1 & 0 & 1
- 1 & 3 & 2
\end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r r }
- 5 & - 3 & 0
1 & - 8 & 1
- 1 & 3 & - 6
\end{array} \right)$$