A \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues 2, 1, 3, with corresponding eigenvectors
$$\left( \begin{array} { l }
1
1
0
\end{array} \right) , \quad \left( \begin{array} { r }
- 1
0
b
\end{array} \right) , \quad \left( \begin{array} { r }
0
1
- 1
\end{array} \right)$$
respectively, where \(b\) is a positive constant.
- Find \(\mathbf { A }\) in terms of \(b\).
- Find \(\mathbf { A } ^ { - 1 } \left( \begin{array} { r } 0
2
- 2 \end{array} \right)\). - It is given that
$$\mathbf { A } ^ { n } \left( \begin{array} { l }
1
1
0
\end{array} \right) = \left( \begin{array} { l }
4
4
0
\end{array} \right) \quad \text { and } \quad \mathbf { A } ^ { n } \left( \begin{array} { r }
- 1
0
b
\end{array} \right) = \left( \begin{array} { c }
- 1
0
b ^ { - 1 }
\end{array} \right) .$$
Find the values of \(n\) and \(b\).