Do you mean matrix inversion, or solving a linear system? The inverse of a sparse matrix is typically nearly dense, so the inverse of your 65000x65000 matrix would not fit in memory.
If you are solving a linear system A*x = b, backslash is a good start. The iterative solvers can be more effective, although it's likely that you will need to find a good preconditioner for your system matrix (maybe a simplified version of your FEM problem could work?).
