8. $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 2 & 5
0 & 3 & p
- 6 & 6 & - 4 \end{array} \right) \quad \text { where } p \text { is a constant }$$ Given that \(\left( \begin{array} { r } 2
1
- 2 \end{array} \right)\) is an eigenvector for \(\mathbf { A }\)

1. 1. determine the eigenvalue corresponding to this eigenvector
   2. hence show that \(p = 2\)
   3. determine the remaining eigenvalues and corresponding eigenvectors of \(\mathbf { A }\)
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\)
3. 1. Solve the differential equation \(\dot { u } = k u\), where \(k\) is a constant. With respect to a fixed origin \(O\), the velocity of a particle moving through space is modelled by $$\left( \begin{array} { c } \dot { x }
      \dot { y }
      \dot { z } \end{array} \right) = \mathbf { A } \left( \begin{array} { l } x
      y
      z \end{array} \right)$$ By considering \(\left( \begin{array} { c } u
      v
      w \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } x
      y
      z \end{array} \right)\) so that \(\left( \begin{array} { c } \dot { u }
      \dot { v }
      \dot { w } \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } \dot { x }
      \dot { y }
      \dot { z } \end{array} \right)\)
   2. determine a general solution for the displacement of the particle.