Singularity of sparse Bernoulli matrices

Let Mn be an n×n random matrix with independent and identically distributed Bernoulli(p) entries. We show that there is a universal constant C≥1 such that, whenever p and n satisfy Clogn∕n≤p≤C−1, P{Mn is singular}=(1+on(1))P{Mn contains a zero row or column}=(2+on(1))n(1−p)n, where on(1) denotes a quantity which converges to zero as n→∞. We provide the corresponding upper and lower bounds on the smallest singular value of Mn as well.