A determinant of exactly zero means a matrix is singular, but numerical code tests whether it is "close to zero". Why is the exact test unusable in practice?

C. Because floating-point arithmetic rarely produces an exact zero.

Rounding means a genuinely singular matrix computes to something like 1e-17 rather than 0, so an exact test declares it invertible and the code divides by that. A tolerance is necessary, and the determinant is a poor basis for one, scaling a 10×10 matrix by 2 multiplies it by 1024. Libraries use the condition number, which is scale-invariant and says how many digits to trust.