What I'm trying to say is that occam's razor seems to suggest that there is a better way to do matrix inversions which we simply haven't discovered yet.

Occam's razor doesn't suggest that. Such a method is an additional assumed entity that adds nothing to our ability to explain what is observed, and is thus the kind of thing Occam's razor calls one to reject, rather than suggests.

Now, belief in the existence of such a method might be a means of satisfying an aesthetic preference for a computationally convenient universe, but that's something very different than Occam's razor.