Wait, no, LU decomposition is not stable unless you're doing pivoting. And if you're doing pivoting, then it very much stops looking like an "exact formula", because it'll get filled up by ifs and variable renamings, etc.

There's plenty of cases in graphics and simulation where solving even a 4x4 matrix needs careful numeric consideration (see this paper about tetrahedral mesh simplification, section III.D, page 7: http://www.sci.utah.edu/~hvo/papers/tetstream.pdf)

If you care about accuracy at all, do not use the straightforward "exact formulas" like Cramer's rule. That's just asking for trouble.

And for numerical linear algebra, I'd go start with Strang and chapter 9: http://math.mit.edu/linearalgebra/