Unless there is something I'm missing here, it seems to me this is a
classic problem that has been explored in the literature using several
approaches.
Depending on the sparsity of the entries, (sparse as polynomials)  or
sparsity of the matrix (as zero entries) and size of coefficients, it
could be
possible to expand by minors, use a modular method, use one of several
versions of Gaussian elimination (eliminating some divisions that
might seem standard), or some graph-based method.

The answer might be so huge that it won't fit in memory, and that
certainly bears some analysis before you try to make it "faster".
RJF


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