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AFAICT, the computations go just fine. One can print any element of the inverse matrix (a very large SR expression).

However, the resultant expressions are way to large for Mathjax (or even for \LaTeX) to cope. Therefore, show (or view) fails spectacularly.

One possible solution is to change ring to QQbar (which is still an exact field). That way, you still can do exact computations and print your results.

Another could be to change ring to a suitably choose cyclotomic field (left as an exercise to the reader...).