9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 8 & - 7 & - 12
- 10 & 5 & 15
- 9 & 6 & 6 \end{array} \right)\) and \(\mathbf { A } ^ { - 1 } = k \left( \begin{array} { r r r } 4 & 2 & 3
5 & 4 & 0
1 & - 1 & 2 \end{array} \right)\).
- Find the exact value of \(k\).
- Using your answer to part (i), solve the following simultaneous equations.
$$\begin{aligned}
8 x - 7 y - 12 z & = 14
- 10 x + 5 y + 15 z & = - 25
- 9 x + 6 y + 6 z & = 3
\end{aligned}$$
You are also given that \(\mathbf { B } = \left( \begin{array} { r r r } - 7 & 5 & 15
a & - 8 & - 21
2 & - 1 & - 3 \end{array} \right)\) and \(\mathbf { B } ^ { - 1 } = \frac { 1 } { 3 } \left( \begin{array} { r r r } 1 & 0 & 5
- 4 & - 3 & 1
2 & 1 & b \end{array} \right)\). - Find the values of \(a\) and \(b\).
- Write down an expression for \(( \mathbf { A B } ) ^ { - 1 }\) in terms of \(\mathbf { A } ^ { - 1 }\) and \(\mathbf { B } ^ { - 1 }\). Hence find \(( \mathbf { A B } ) ^ { - 1 }\).