\(\mathbf { 9 }\) You are given that \(\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 5
3 & a & 1
1 & - 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 2 a + 1 & 3 & 1 + 5 a
- 5 & 1 & - 13
- 3 - a & - 1 & - 2 a - 3 \end{array} \right)\).
- Show that \(\mathbf { A B } = ( 8 + a ) \mathbf { I }\).
- State the value of \(a\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Write down \(\mathbf { A } ^ { - 1 }\) in terms of \(a\), when \(\mathbf { A } ^ { - 1 }\) exists.
- Use \(\mathbf { A } ^ { - 1 }\) to solve the following simultaneous equations.
$$\begin{aligned}
- 2 x + y - 5 z & = - 55
3 x + 4 y + z & = - 9
x - y + 2 z & = 26
\end{aligned}$$ - What can you say about the solutions of the following simultaneous equations?
$$\begin{aligned}
- 2 x + y - 5 z & = p
3 x - 8 y + z & = q
x - y + 2 z & = r
\end{aligned}$$