Fast method to check if a Matrix is singular? (non-invertible, det = 0)

I agree with Gaussian elimination. http://math.nist.gov/javanumerics/jama/doc/Jama/LUDecomposition.html documents LU decomposition - after constructing the LU decomposition from a matrix you can call a method on it to get the determinant. My guess is that it is at least worth timing this to compare it with any more specialised scheme.

Best way is to compute the condition number via SVD and check if it is greater than 1 / epsilon, where epsilon is the machine precision.

If you allow false negatives (ie. a matrix is defective, but your algorithm may not detect it), you can use the max(a_ii) / min(a_ii) formula from the Wikipedia article as a proxy for the condition number, but you have to compute the QR decomposition first (the formula applies to triangular matrices): A = QR with R orthogonal, then cond(A) = cond(Q). There are also techniques to compute the condition number of Q with O(N) operations, but there are more complex.