I'm learning about the Bareiss algorithm to compute determinants over integral domains from Chee Yap, Fundamental Problems of Algorithmic Algebra (see also this question). The naive version of the algorithm suffers from zero divisions, i.e. it will not work if there are some leading principal minors that are 0.
As mentioned briefly in the text - and also more extensively in this paper (p. 34) - the workaround is to exchange rows (or columns, I guess this should be equivalent): If I'm in the k-th outermost step and the (k,k) entry is zero, I find a row i with i > k, such that the (i,k) entry is non-zero (and then multiply the determinant by -1).
Now my question is: What if I can't find any such row?
There is some hint later on in the Yap text about this meaning that the matrix has dependent rows (so the determinant would be 0). But I don't have a proof for this.
Additionally, it would be useful if somebody knew a concrete example of a matrix that exhibits this behaviour. The only one I can think of is the zero matrix which, trivially, does have determinant 0.