Complex Numbers

For two unimodular complex numbers $$z_1$$ and $$z_2$$, $$\begin{bmatrix} \overline z_1& -z_2 \\ \overline z_2 & z_1\end{bmatrix}^{-1}$$$$\begin{bmatrix} z_1& z_2 \\ -\overline z_2 & \overline z_1\end{bmatrix}^{-1}$$ is equal to
For two unimodular complex numbers $$z_1$$ and $$z_2$$, $$\begin{bmatrix} \overline z_1& -z_2 \\ \overline z_2 & z_1\end{bmatrix}^{-1}$$$$\begin{bmatrix} z_1& z_2 \\ -\overline z_2 & \overline z_1\end{bmatrix}^{-1}$$ is equal to
$$\dfrac { 5+2i }{ 1-i } +\dfrac { 5-2i }{ 1+i } $$
Let $$z_1$$ = 18 + 83i, $$z_2$$ = 18 + 39i, ana $$z_3 $$= 78 + 99i. where i = $$\sqrt-1$$. Let z be a unique comlpex number with the properties that $$\dfrac{z_3 - z_1}{z_2 - z_1}$$ $$\cdot$$ $$\dfrac{z - z_2}{z - z_3}$$ is a real number and the imaginary part of the size z is the greatest possible.
$$\dfrac { 5+2i }{ 1-i } +\dfrac { 5-2i }{ 1+i } $$