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

Terence Tao, Van Vu

Published 2004-11-04, updated 2008-06-30Version 5

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 $\sqrt {n!} \exp(O(\sqrt(n log n)))$. 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.