Find the inverse of the matrix
$$\left( \begin{array} { r r r }
1 & 1 & a
2 & - 1 & 2
3 & - 2 & 2
\end{array} \right)$$
where \(a \neq 4\).
Show that when \(a = - 1\) the inverse is
$$\frac { 1 } { 5 } \left( \begin{array} { r r r }
2 & 0 & 1
2 & 5 & - 4
- 1 & 5 & - 3
\end{array} \right)$$
Solve, in terms of \(b\), the following system of equations.
$$\begin{aligned}
x + y - z & = - 2
2 x - y + 2 z & = b
3 x - 2 y + 2 z & = 1
\end{aligned}$$
Find the value of \(b\) for which the equations
$$\begin{aligned}
x + y + 4 z & = - 2
2 x - y + 2 z & = b
3 x - 2 y + 2 z & = 1
\end{aligned}$$
have solutions. Give a geometrical interpretation of the solutions in this case.
Section B (18 marks)