Wikipedia gives this form of the stereographic Projection from $S^{2} \rightarrow \hat{\mathbb C}$:$ (1) : z=\frac{x_{1}+ix_{2}}{1-x_{3}}$ and for the inverse projections the points are supposedly:$(2): x_{1}=\frac{\overline{z}+z}{z\overline{z}+1}, x_{2}=\frac{z-\overline{z}}{i(z\overline{z}+1)}, x_{3}=\frac{z\overline{z}-1}{z\overline{z}+1}$
How does he go from $(1)$ to $(2)$ ? I tried calculating the inverse matrix and reading the coefficients from it, looking at $\overline{z}, z$ in $(1)$ and solving it for $x_{1},x_{2},x_{3}$ but I don't get anything of this form
(f.e. I get: $\displaystyle x_{1}=\frac{z(1-x_{3})-2ix_{2}}{\overline{z}(1-x_{3})})$