# Singularity of random Bernoulli matrices

## TL;DR

This paper proves that the probability of a random Bernoulli matrix being singular asymptotically approaches ^n, resolving a longstanding open problem in random matrix theory.

## Contribution

It establishes the exact asymptotic probability of singularity for random Bernoulli matrices, a problem that remained open for decades.

## Key findings

- Probability of singularity approaches (1/2)^n as n grows.
- Provides a rigorous proof settling the old conjecture.
- Includes some generalizations beyond the basic model.

## Abstract

For each $n$, let $M_n$ be an $n\times n$ random matrix with independent $\pm 1$ entries. We show that ${\mathbb P}\{\mbox{$M_n$ is singular}\}=(1/2+o_n(1))^n$, which settles an old problem. Some generalizations are considered.