Find matrix A given eigenvalues and eigenvectors

6 The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 6 & 6 \\ 0 & 2 & 6 \\ 0 & 0 & - 3 \end{array} \right) .$$

1. Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
2. Find the matrix \(\mathbf { A }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { l l l } 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{array} \right) .$$
3. State the eigenvalues and corresponding eigenvectors of \(\mathbf { A } ^ { 3 }\).

6 The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & - 1 \end{array} \right) .$$

1. State the eigenvalues of \(\mathbf { P }\).
2. Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
   The \(3 \times 3\) matrix \(\mathbf { A }\) has distinct non-zero eigenvalues \(a , \frac { 1 } { 2 } , 2\) with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 2 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } 1 \\ 1 \\ - 1 \end{array} \right) ,$$ respectively.
3. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).

3 Find a matrix \(\mathbf { A }\) whose eigenvalues are \(- 1,1,2\) and for which corresponding eigenvectors are $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 0 \\ 1 \\ 1 \end{array} \right) ,$$ respectively.