In general there are no shortcuts, and when the matrix A is not small (e.g. m x n with m,n << 100), you don't want to invert A, especially if you are only interested in x (from A x = b). Instead, use numerical minimization schemes like conjugated gradients and variants thereof or Quasi-Newton methods (BFGS). Combined with preconditioning as well as regularization or denoising this usually yields good results. Compressive sampling methods (total variation regularization) have received lots of attention in the last ten years.
This comment seems to be a diversion from the question asked. The main point of OP is that solving A x = b is commonly done with factorizations of A, not by explicit computation of inv(A).
Resorting to a numerical minimization (like MINRES) would be unusual unless the dimension is much bigger than hundreds. Certainly it's not relevant in a graphics computation with dimension of 3, 4, or 6. My laptop inverts a 4Kx4K general matrix in 1.5s. It solves a 4K linear system in 0.5s.
