[mturmon]

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.