Under what conditions the function y =(ax+b/cx+d) , (ad - bc) is not equal to 0. is it’s own inverse?

First inverse the function :
x = ay + b / cy + d
xcy + dx = ay + b
y = b - dx/ cx - a
It should be equal to original function
b - dx/ cx - a = ax + b/cx + d
bcx + bd - dcx^2 - d^2x = acx^2 + bcx - a^2x - ab

By this, we get a = -d should be the other condition.
Thanks
Bharat
askiitians faculty
IIT Delhi