Use a library instead, and use the most specific methods that are available: if there are specialized routines to solve Ax=b, use them, don't use the matrix inversion methods (which are probably also there), or even the LU decomposition ones, even if Real Math tells you that the result is equivalent - in numerical computation, mathematical equivalence is an entirely different entity than what "real" mathematicians concern themselves with.
Seriously. I don't care if you're a math PhD, I don't even care if you know a decent amount about numerical methods. Whatever code you're going to put together to do the job will be inferior to what a mature library can do, simply because the library has been put through many iterations and has had a lot more time to get patched up to handle all those special cases (usually related to finite precision or discretization) that Real Math doesn't have to worry about. And it will probably be vastly more optimized than anything you'll put together, because it's spent years of heavy use in some critical research environments.
The only time you should be writing your own math code (beyond simple arithmetic) is if no mature libraries are available to you. In which case you should allocate at the very least twice the amount of time it will take to research and implement the methods, and probably a lot more, because you're going to have a lot of fiddling to do.