try using a software named MATLAB: it has readymade methods of finding matrix inverse if you don't need to show to your professor/teacher how you got the answer matrix: X
WELL, THERE IS A LOT OF WAYS TO SOLVE YOUR PROBLEM!
First Gauss-Seidel, then Jacobi.
Now Moore-Penrose.
I just remembered one more: Gauss-Jordan elimination method of finding a MATRIX INVERSE
http://mathworld.wolfram.com/Gauss-JordanElimination.html
P.S.: ARE YOU AN ENGINEERING STUDENT?
Singularity of a matrix means that it becomes discontinuous at that point: take for example a black hole: it has a radius zero, which means that its gravitational force becomes undefined then; the same case here is when the adjoint of the co-efficient matrix we got is to be divided by the...