3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 7 & 3
- 4 & - 1 \end{array} \right)\).
- Find the eigenvalues, and corresponding eigenvectors, of the matrix \(\mathbf { M }\).
- Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\).
- Given that \(\mathbf { M } ^ { n } = \left( \begin{array} { l l } a & b
c & d \end{array} \right)\), show that \(a = - \frac { 1 } { 2 } + \frac { 3 } { 2 } \times 5 ^ { n }\), and find similar expressions for \(b , c\) and \(d\).
Section B (18 marks)