http://mathworld.wolfram.c

http://en.wikipedia.org/wi

It works, you can do it by hand, but it is not likely to be much fun.

Even in Linear Algebra courses, they don't make you work with anything over 4x4.

There are solvers on line that will do up to 30x30.

http://www.bluebit.gr/matr

Matlab is probably the software of choice for this sort of thing.