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## Möbius transformations representable as 2x2 matrices

Although there may be some question of whether the implicit or the explicit form of one-to-one and invertible mappings should be called Möbius transformations, that is the name given to this class of transformations. Mappings often form a group. Since the associative law for their composition is a given, the items to be checked are closure, invertibility and the existence of the identity.

To check closure, consider the composite of two maps. First, suppose we have

 t = (29) w = (30)

which gives, on substituting,
 t = (31)

and, on simplifying,
 t = (32)

which mimics the matrix product

From this it is apparent that the unimodular matrices form a representation of the group of Möbius transformations; in particular the identity is represented by the unit matrix,
 z = (33)

and the inverse transformation by the inverse matrix.

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