3 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } 3 & 3 & 0
0 & 2 & 2
1 & 3 & 4 \end{array} \right)\).
- Determine the characteristic equation of \(\mathbf { A }\).
- Hence verify that the eigenvalues of \(\mathbf { A }\) are 1, 2 and 6 .
- For each eigenvalue of \(\mathbf { A }\) determine an associated eigenvector.
- Use the results of parts (b) and (c) to find \(\mathbf { A } ^ { n }\) as a single matrix, where \(n\) is a positive integer.