10 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { l l l }
6 & - 8 & 7
7 & - 9 & 7
6 & - 6 & 5
\end{array} \right)$$
- Given that \(\left( \begin{array} { l } 1
1
0 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\), find the corresponding eigenvalue. - Given also that - 1 is an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector.
- It is given that the determinant of \(\mathbf { A }\) is equal to the product of the eigenvalues of \(\mathbf { A }\). Use this result to find the third eigenvalue of \(\mathbf { A }\), and find also a corresponding eigenvector.
- Write down matrices \(\mathbf { P }\) and \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\), where \(\mathbf { D }\) is a diagonal matrix, and hence find the matrix \(\mathbf { A } ^ { n }\) in terms of \(n\), where \(n\) is a positive integer.