Never invert a square matric explicitly: it is highly numerically unstable. A much
better method is to find two triangular matrixes L and U (lower and upper triangular
respectively where L has just ones (1) on its diagonal such that L.U == P.A
where P is a row permutation matrix, Given L, U and P it is extremely easy
to solve the linear system A.x == b.
P.A.x == P.b (swap the elements in b)
L.U.x == P.b
L.y == P.b (by using a simple backward substitution)
U.x == y (by using a simple forward substitution).
Calculate x for the unit vectors b and voila.
Google for "LUP decomposition" for the gory details.
kind regards,
Jos