10 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 3 & 4 & - 1
1 & - 1 & k
- 2 & 7 & - 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r c } 11 & - 5 & - 7
1 & 11 & 5 + k
- 5 & 29 & 7 \end{array} \right)\) and that \(\mathbf { A B }\) is of the form \(\mathbf { A B } = \left( \begin{array} { c c c } 42 & \alpha & 4 k - 8
10 - 5 k & - 16 + 29 k & - 12 + 6 k
0 & 0 & \beta \end{array} \right)\).

1. Show that \(\alpha = 0\) and \(\beta = 28 + 7 k\).
2. Find \(\mathbf { A B }\) when \(k = 2\).
3. For the case when \(k = 2\) write down the matrix \(\mathbf { A } ^ { - 1 }\).
4. Use the result from part (iii) to solve the following simultaneous equations. $$\begin{aligned} 3 x + 4 y - z & = 1
   x - y + 2 z & = - 9
   - 2 x + 7 y - 3 z & = 26 \end{aligned}$$ \footnotetext{OCR
   RECOGNISING ACHIEVEMENT