6. $$\mathbf { M } = \left( \begin{array} { r r r } p & - 2 & 0
- 2 & 6 & - 2
0 & - 2 & q \end{array} \right)$$ where \(p\) and \(q\) are constants.
Given that \(\left( \begin{array} { r } 2
- 2
1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),

1. find the eigenvalue corresponding to this eigenvector,
2. find the value of \(p\) and the value of \(q\). Given that 6 is another eigenvalue of \(\mathbf { M }\),
3. find a corresponding eigenvector. Given that \(\left( \begin{array} { l } 1
   2
   2 \end{array} \right)\) is a third eigenvector of \(\mathbf { M }\) with eigenvalue 3
4. find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$