5 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { l l } a & 0
2 & 3 \end{array} \right)\) where \(a\) is a constant and \(a \neq 3\).

1. Given that the acute angle between the directions of the eigenvectors of \(\mathbf { P }\) is \(\frac { 1 } { 4 } \pi\) radians, determine the possible values of \(a\).
2. You are given instead that \(\mathbf { P }\) satisfies the matrix equation \(\mathbf { I } = \mathbf { P } ^ { 2 } + r \mathbf { P }\) for some rational number \(r\).
   1. Use the Cayley-Hamilton theorem to determine the value of \(a\) and the corresponding value of \(r\).
   2. Hence show that \(\mathbf { P } ^ { 4 } = \mathbf { s } \mathbf { + t } \mathbf { t } \mathbf { P }\) where \(s\) and \(t\) are rational numbers to be determined. You should not calculate \(\mathbf { P } ^ { 4 }\).