In Algol 68 there is this which is a plain inversion routine.
There is this more exotic routine based on decomposing the matrix into blocks:
copied from here. The first of these inverts all invertible matrices. The 2nd, as it stands, certainly does not. There are probably ill-conditioned matrices that other routines could handle better.
The inversion routine in this Scheme matrix suite inverts all invertible matrices M and returns a non-zero vector x such that Mx=0 as it reports a singular matrix—a constructive proof of its singularity and a value sought in some eigenvector algorithms. The code can work with exact rational complex numbers, or even finite fields. Some of the logic is described here.
Some C code with calling logic included. It unifies solution and inversion.
This OCaml module includes matrix inversion code.