7 The matrix \(\mathbf { P }\) is given by
$$\mathbf { P } = \left( \begin{array} { r r r }
1 & 4 & 2
0 & - 1 & 1
0 & 0 & 2
\end{array} \right) .$$
- State the eigenvalues of \(\mathbf { P }\).
- Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
The \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues \(b , - 1,1\) with corresponding eigenvectors
$$\left( \begin{array} { l }
1
0
0
\end{array} \right) , \quad \left( \begin{array} { r }
4
- 1
0
\end{array} \right) , \quad \left( \begin{array} { l }
2
1
2
\end{array} \right)$$
respectively. - Find \(\mathbf { A }\) in terms of b.