It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf{A}\), with corresponding eigenvector \(\mathbf{e}\). 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

\(\mathbf{A}=\begin{pmatrix}2&0&1\\-1&2&3\\1&0&2\end{pmatrix}.\)

Find a corresponding eigenvector.

It is also given that \(\begin{pmatrix}0\\1\\0\end{pmatrix}\) and \(\begin{pmatrix}1\\2\\1\end{pmatrix}\) are eigenvectors of \(\mathbf{A}\). Find the corresponding eigenvalues.

Hence find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\left(\mathbf{A}+\mathbf{A}^{-1}\right)^3=\mathbf{PDP}^{-1}\).