This site presents a method for inverting a matrix that may be singular or non-square. Included on this site is a demonstration page where you may enter a matrix and have the script calculate the inverse. The information herein has been around for decades, and nothing I make is ever new or unique, but it was good enough for a successful dissertation. I've worded this presentation in a way that skirts discussions about engenvectors, eigenvalues, or other topics that may scare off anyone trying to learn an easy way to solve a linear algebra problem.
I misplaced the original matrix inversion pages, so I've had to recreate these from scratch. I refused to pay US$65 semi-annually just to have this site hosted on a Java server so my servlet could perform calculations and return the results to the page.
The instructions begin on this page. If you haven't been through the instructions, I'd recommend doing so before jumping off to the demo page, or grabbing the source and just blindly using the code. Note also that the sources and script do some additional shuffling of the data to avoid numerical instability (I choose to never divide a large number by a small one, for instance.)