The square matrix \(\mathbf{A}\) has \(\lambda\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector. Show that \(\mathbf{e}\) is an eigenvector of \(\mathbf{A}^{2}\) and state the corresponding eigenvalue.

Find the eigenvalues of the matrix \(\mathbf{B}\), where
\(\mathbf{B}=\left(\begin{array}{lll} 1 & 3 & 0 \\ 2 & 0 & 2 \\ 1 & 1 & 2 \end{array}\right) .\)

Find the eigenvalues of \(\mathbf{B}^{4}+2 \mathbf{B}^{2}+3 \mathbf{I}\), where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix.