6 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by
$$| z - 2 \mathrm { i } | = 2 \quad \text { and } \quad | z + 1 | = | z + \mathrm { i } |$$
respectively.
- Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
- Hence write down the complex numbers represented by the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
\(7 \quad\) The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r r } a & 1 & 3
2 & 1 & - 1
0 & 1 & 2 \end{array} \right)\). - Given that \(\mathbf { B }\) is singular, show that \(a = - \frac { 2 } { 3 }\).
- Given instead that \(\mathbf { B }\) is non-singular, find the inverse matrix \(\mathbf { B } ^ { - 1 }\).
- Hence, or otherwise, solve the equations
$$\begin{aligned}
- x + y + 3 z & = 1
2 x + y - z & = 4
y + 2 z & = - 1
\end{aligned}$$