If det(A) v det(B)=0, then you get zero times another scalar, which is zero.
Hence, det(A) and det(B) ≠ 0.
If AB is singular, then,
If neither scalars det(A) or det(B)≠0, then neither one of them could be singular, sinceall nonsingular matricies have nonzero determinants; so, det(A)det(B) wouldn't be equal to zero.
So, A is singular, or B is singular.
Recall that for all scalars a b in R ab=0 iff (a=0)v(b=0)