IMC 2025 · Problem 3

Denote by $\mathcal{S}$ the set of all real symmetric $2025 \times 2025$ matrices of rank $1$ whose entries take values $-1$ or $+1$. Let $A, B \in \mathcal{S}$ be matrices chosen independently uniformly at random. Find the probability that $A$ and $B$ commute, i.e. $AB = BA$.