What do you mean with inverting a matrix? In the usual definition it is well studied and understood https://en.m.wikipedia.org/wiki/Invertible_matrix#Methods_of...
I know I may be asking to make what may be a fundamentally-hard problem, easy... I'm just saying that I don't get the same, eh, "buzz" I get when I see a really cool solution for something (which I've often gotten from math, physics, programming, etc.). This is very hard to explain as it's probably irrational and thus I can't make a strong rational argument for it lol (admittedly!)
I really love linear algebra, btw, so this should be right up my alley.