4 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
- 11 & 1 & 8
0 & - 2 & 0
- 16 & 1 & 13
\end{array} \right)$$
- Show that \(\left( \begin{array} { l } 1
1
1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and state the corresponding eigenvalue. - Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 19 \lambda - 30 = 0\) and hence find the other eigenvalues of \(\mathbf { A }\).
\includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-08_2717_35_106_2015}
\includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-09_2726_33_97_22} - Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).