$$A = \left( \begin{array} { r r }
1 & - 2
1 & 4
\end{array} \right)$$
- Show that the characteristic equation for \(\mathbf { A }\) is \(\lambda ^ { 2 } - 5 \lambda + 6 = 0\)
- Use the Cayley-Hamilton theorem to find integers \(p\) and \(q\) such that
$$\mathbf { A } ^ { 3 } = p \mathbf { A } + q \mathbf { I }$$
(ii) Given that the \(2 \times 2\) matrix \(\mathbf { M }\) has eigenvalues \(- 1 + \mathrm { i }\) and \(- 1 - \mathrm { i }\), with eigenvectors \(\binom { 1 } { 2 - \mathrm { i } }\) and \(\binom { 1 } { 2 + \mathrm { i } }\) respectively, find the matrix \(\mathbf { M }\).