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I had a similar problem for a 24X24 symbolic matrix. Mathematica could not do it and I ended up using Fermats "Redrowech[]". It took only 30 seconds.

The problem is, Fermat's interface is far from optimal and the coding in it is, lets say cumbersome. Here is how I approached it:

1. Create Fermat input with Mathematica (define the array and the variables)
2. Append the created file to "ferstart.txt"
3. Call the "gate to fermat" interface from Mathematica to Fermat (Link to Fermat FLINK)
4. Run from Mathematica FEval["Redrowech[yourMatrix]"] and save the result

The routine I used is tailored towards my particular matrices and I am therefore reluctant to share it. But if you have troubles I might be able to help.