The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} -2 & 0 & 0 \\ 0 & 7 & 9 \\ 4 & 1 & 7 \end{array}\right) .\)
(a) Show that the characteristic equation of \(\mathbf{A}\) is \(\lambda^{3}-12 \lambda^{2}+12 \lambda+80=0\) and find the eigenvalues of A.

(b) Use the characteristic equation of \(\mathbf{A}\) to show that
\(\mathbf{A}^{4}=p \mathbf{A}^{2}+q \mathbf{A}+r \mathbf{I},\)
where \(p, q\) and \(r\) are integers to be determined.
(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A}-3 \mathbf{I})^{4}=\mathbf{P D P}^{-1}\).