> On the other hand, open source tools are catching up.

SymPy is doing nicely, but it's decades behind Mathematica when it comes to symbolic computation.

By the way, if you want to invert a symbolic matrix, chances are that generic Gaussian elimination working with symbolic expressions isn't the best algorithm (it might even give wrong answers if zero testing is done incorrectly), and compiled code isn't going to change that. The state of the art in symbolic linear algebra uses specialized algorithms like evaluation-interpolation, modular computation, and all kinds of low level optimizations.