1. $$\mathbf { A } = \left( \begin{array} { r r } 4 & - 1
7 & 2
- 5 & 8 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 3 & 2
- 1 & 6 & 5 \end{array} \right) \quad \mathbf { C } = \left( \begin{array} { r r r } - 5 & 2 & 1
4 & 3 & 8
- 6 & 11 & 2 \end{array} \right)$$ Given that \(\mathbf { I }\) is the \(3 \times 3\) identity matrix,

1. 1. show that there is an integer \(k\) for which $$\mathbf { A B } - 3 \mathbf { C } + k \mathbf { I } = \mathbf { 0 }$$ stating the value of \(k\)
   2. explain why there can be no constant \(m\) such that $$\mathbf { B A } - 3 \mathbf { C } + m \mathbf { I } = \mathbf { 0 }$$
2. 1. Show how the matrix \(\mathbf { C }\) can be used to solve the simultaneous equations $$\begin{aligned} - 5 x + 2 y + z & = - 14
      4 x + 3 y + 8 z & = 3
      - 6 x + 11 y + 2 z & = 7 \end{aligned}$$
   2. Hence use your calculator to solve these equations.