The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 5 & -\frac{22}{3} & 8 \\ 0 & -6 & 0 \\ 0 & 0 & 1 \end{array}\right)\)
(a) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{2}=\mathbf{P D P}^{-1}\).
(b) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{3}\).