The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} -6 & 2 & 13 \\ 0 & -2 & 5 \\ 0 & 0 & 8 \end{array}\right) .\)
(a) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{-1}=\mathbf{P D P} \mathbf{P}^{-1}\).
(b) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\).