6 The matrix $\mathbf { A }$ is given by

$$A = \left( \begin{array} { r r r } 
2 & - 3 & - 7 \\
0 & 5 & 7 \\
0 & 0 & - 2
\end{array} \right) .$$

(a) Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }$.\\

(b) Use the characteristic equation of $\mathbf { A }$ to show that

$$\mathbf { A } ^ { 4 } = a \mathbf { A } ^ { 2 } + b \mathbf { I } ,$$

where $a$ and $b$ are integers to be determined.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2022 Q6}}