Just because you want to do something does not mean it is possible, at least computationally possible, with the hardware available. A symbolic matrix of that size will result in an immense number of terms in any element of the inverse. Trying to do the job by hand, so to speak, thus using determinants, Gaussian elimination, etc., will be as nasty.
Sorry. But you need to rethink your needs. You can't have what you want in this case. Of course, if you are willing to wait a sufficient number of years for Moore's law to catch up to you, then no problem! No, I don't know how long that will take. And there are some valid reasons to wonder if Moore's law might at some point reach a fundamental impasse, at least based on the current hardware trends.
