video

given $a,b,c,d$ real, such that $ad-bc > 0$, a Moebius transformation is:

$z \rightarrow \frac{az+b}{cz+d} = w$.

They are invertible in the upper half-plane.

The inverse is $w \rightarrow \frac{dw-b}{-cw+a}$, another Moebius transformation, which can be get as a 2x2 inverse matrix

They are Isometryes of the upper half-plane model of Hyperbolic geometry (see here for another proof)