Given that 5 is an eigenvalue of the matrix
\(\mathbf{A}=\left(\begin{array}{rrr} 5 & -3 & 0 \\ 1 & 2 & 1 \\ -1 & 3 & 4 \end{array}\right),\)
find a corresponding eigenvector.

Hence find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf{A}+\mathbf{A}^{2}\).