Prove that any Möbius transformation can be written in a form with determinant 1, and that this form is unique up to sign. How

Question

Prove that any Möbius transformation can be written in a form with determinant 1, and that this form is unique up to sign.
How does the determinant of T(z) = (az+b)/(cz + d) change if we multiply top and bottom of the map by some constant k?

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