8 It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf { A }\), with corresponding eigenvector
e. Show that \(\lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } ^ { - 1 }\) for which \(\mathbf { e }\) is a corresponding eigenvector. Deduce that \(\lambda + \lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } + \mathbf { A } ^ { - 1 }\). It is given that 1 is an eigenvalue of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1
- 1 & 2 & 3
1 & 0 & 2 \end{array} \right)$$ Find a corresponding eigenvector. It is also given that \(\left( \begin{array} { l } 0
1
0 \end{array} \right)\) and \(\left( \begin{array} { l } 1
2
1 \end{array} \right)\) are eigenvectors of the matrix \(\mathbf { A }\). Find the corresponding eigenvalues.
Hence find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\left( \mathbf { A } + \mathbf { A } ^ { - 1 } \right) ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$