On random $\pm 1$ matrices: Singularity and Determinant

Abstract

This papers contains two results concerning random $n\times n$ Bernoulli matrices. First, we show that with probability tending to one the determinant has absolute value $\sqrt{n!}\exp (O(\sqrt{(}nlogn)))$. Next, we prove a new upper bound $.939^n$ on the probability that the matrix is singular. We also give some generalizations to other random matrix models.

25 pages, no figures. Slight numerical corrections to Lemma 2.2