The matrix \(\mathbf{P}\) is given by
\(\mathbf{P}=\left(\begin{array}{rrr} 1 & -1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & -1 \end{array}\right) .\)
(a) State the eigenvalues of \(\mathbf{P}\).

(b) Use the characteristic equation of \(\mathbf{P}\) to find \(\mathbf{P}^{-1}\).

The \(3 \times 3\) matrix \(\mathbf{A}\) has distinct non-zero eigenvalues \(a, \frac{1}{2}, 2\) with corresponding eigenvectors
\(\left(\begin{array}{l} 1 \\ 0 \\ \end{array}\right), \quad\left(\begin{array}{r} -1 \\ 2 \\ \end{array}\right), \quad\left(\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right),\)
respectively.
(c) Find \(\mathbf{A}^{-1}\) in terms of \(a\).