9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & 3 & - 1
- 1 & \alpha & - 1
- 2 & - 1 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c c } 3 \alpha - 1 & - 8 & \alpha - 3
5 & 1 & 2
2 \alpha + 1 & - 5 & \alpha + 3 \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } \gamma & 0 & 0
\beta & \gamma & 0
0 & 0 & \gamma \end{array} \right)\).

1. Show that \(\beta = 0\).
2. Find \(\gamma\) in terms of \(\alpha\).
3. Write down \(\mathbf { A } ^ { - 1 }\) for the case when \(\alpha = 2\). State the value of \(\alpha\) for which \(\mathbf { A } ^ { - 1 }\) does not exist.
4. Use your answer to part (iii) to solve the following simultaneous equations. $$\begin{aligned} x + 3 y - z & = 25
   - x + 2 y - z & = 11
   - 2 x - y + 3 z & = - 23 \end{aligned}$$