Not sure about inversion, but if I remember correctly, LAPACK can do for factorizations such as LU in place. You can the refuse the factorization to solve for many RHS.
The normal way (as far as I know) to do inversion in LAPACK is to do LU (in place), and then do invert (also in place) on the LU factors (so they get eaten up by the invert). I guess it would be possible to do: LU, then do some solves, then do invert with those LU factors, it just seems a little odd.