  Starting from an n-by-n matrix of zeros, choose uniformly random zero entries
and change them to ones, one-at-a-time, until the matrix becomes invertible. We
show that with probability tending to one as n tends to infinity, this occurs
at the very moment the last zero row or zero column disappears. We prove a
related result for random symmetric Bernoulli matrices, and give quantitative
bounds for some related problems. These results extend earlier work by Costello
and Vu [arXiv:math/0606414].
