4. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & - 2 & - 7
3 & k & 2
1 & 1 & 4 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { c c c } 4 k - 2 & 1 & 7 k - 4
- 10 & 3 & - 19
3 - k & - 1 & 6 - k \end{array} \right)$$ where \(k\) is a constant.

1. Determine the value of the constant \(c\) for which $$\mathbf { A B } = ( 3 k + c ) \mathbf { I }$$
2. Hence determine the value of \(k\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Given that \(\mathbf { A } ^ { - 1 }\) does exist,
3. write down \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
4. Use the answer to part (c) to solve the simultaneous equations $$\begin{aligned} - x - 2 y - 7 z & = 10
   3 x + k y + 2 z & = 3
   x + y + 4 z & = 1 \end{aligned}$$ giving the values of \(x , y\) and \(z\) in simplest form in terms of \(k\).