1.
$$\mathbf { A } = \left( \begin{array} { r r }
- 1 & a
3 & 8
\end{array} \right)$$
where \(a\) is a constant.
- Determine, in expanded form in terms of \(a\), the characteristic equation for \(\mathbf { A }\).
- Hence use the Cayley-Hamilton theorem to determine values of \(a\) and \(b\) such that
$$\mathbf { A } ^ { 3 } = \mathbf { A } + b \mathbf { I }$$
where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.