4. $$\mathbf { A } = \left( \begin{array} { r r r } 4 & 2 & 0
2 & p & - 2
0 & - 2 & 2 \end{array} \right) \quad \text { where } p \text { is a constant }$$ Given that \(\left( \begin{array} { r } 2
- 1
2 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\),

1. determine the eigenvalue corresponding to this eigenvector.
2. Hence show that \(p = 3\)
3. Determine
   1. the remaining eigenvalues of \(\mathbf { A }\),
   2. corresponding eigenvectors for these eigenvalues.
4. Hence determine a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { \mathrm { T } }\)