2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/166464Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability 1/2). We show that the probability that $M_n$ is singular is at most $(3/4 +o(1))^n$, improving an earlier estimate of Kahn, Komlós and Szemerédi, as well as earlier work by the authors. The key new ingredient is the applications of Freiman type inverse theorems and other tools from additive combinatorics.30 pages, no figures. This is the final versionCombinatorics15A52text