The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 4 & 1 & -1 \\ -4 & -1 & 4 \\ 0 & -1 & 5 \end{array}\right) .\)

Given that one eigenvector of \(\mathbf{A}\) is \(\left(\begin{array}{r}1 \\ -2 \\ -1\end{array}\right)\), find the corresponding eigenvalue.

Given also that another eigenvalue of \(\mathbf{A}\) is 4, find a corresponding eigenvector.

Given further that \(\left(\begin{array}{r}1 \\ -4 \\ -1\end{array}\right)\) is an eigenvector of \(\mathbf{A}\), with corresponding eigenvalue 1 , find matrices \(\mathbf{P}\) and \(\mathbf{Q}\), together with a diagonal matrix \(\mathbf{D}\), such that \(\mathbf{A}^{5}=\mathbf{P D Q}\).