The problem you describe is equivalent to computing the "rank of a matrix." A nonsingular matrix is a square matrix that has full rank. Because of finite-precision arithmetic, the rank problem is difficult. As Ian points out, there are problems with trying to compare a floating-point computation (determinant, eigenvalue, singular value,...) with zero. Stable rank-detecting algorithms use some variation of Gaussian elimination to determine whether there are linear dependencies in the rows or columns of a matrix. In SAS/IML, the standard way to detect rank is to compute the generalized inverse (which always exists) and then multiply it by the original matrix. If you get the identity matrix, then the original matrix was full rank (nonsingular). Otherwise the original matrix is less than full rank. So to detect singularity, I suggest the following:
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