10 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & - 2 & k
2 & 1 & 2
3 & 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r c c } - 5 & - 2 + 2 k & - 4 - k
8 & - 1 - 3 k & - 2 + 2 k
1 & - 8 & 5 \end{array} \right)\) and that \(\mathbf { A B }\) is of the form \(\mathbf { A B } = \left( \begin{array} { c c c } k - n & 0 & 0
0 & k - n & 0
0 & 0 & k - n \end{array} \right)\).

1. Find the value of \(n\).
2. Write down the inverse matrix \(\mathbf { A } ^ { - 1 }\) and state the condition on \(k\) for this inverse to exist.
3. Using the result from part (ii), or otherwise, solve the following simultaneous equations. $$\begin{aligned} x - 2 y + z = & 1
   2 x + y + 2 z = & 12
   3 x + 2 y - z = & 3 \end{aligned}$$