The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 18 & 5 & -11 \\ 8 & 6 & -4 \\ 32 & 10 & -20 \end{array}\right)\)
(a) Show that the characteristic equation of \(\mathbf{A}\) is \(\lambda^{3}-4 \lambda^{2}-20 \lambda+48=0\) and hence find the eigenvalues of \(\mathbf{A}\).
(b) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{5}=\mathbf{P D P}^{-1}\).