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Please provide a concrete example, perhaps by providing a link to a worksheet or so.

I just tried (using sage-4.7.2):

sage: MS = MatrixSpace(QQ, 50)
sage: A = MS.random_element()
sage: %time B = ~A
CPU times: user 0.23 s, sys: 0.00 s, total: 0.23 s
Wall time: 0.38 s

So, 50x50 over QQ really doesn't seem to be a challenge, and thus it would be interesting to know what you did exactly.