International Mathematics Competition
for University Students
2025

IMC2025: Day 1, Problem 3

Problem 3. Denote by \(\displaystyle \mathcal{S}\) the set of all real symmetric \(\displaystyle 2025\times 2025\) matrices of rank \(\displaystyle 1\) whose entries take values \(\displaystyle -1\) or \(\displaystyle +1\). Let \(\displaystyle A,B\in\mathcal{S}\) be matrices chosen independently uniformly at random. Find the probability that \(\displaystyle A\) and \(\displaystyle B\) commute, i.e. \(\displaystyle AB=BA.\)

Marian Panţiruc, "Gheorghe Asachi" Technical University of Iaşi, Romania

  Solution