IMC 2013 · Problem 1

Day 120th IMC · Blagoevgrad, Bulgaria

Statement

Let $A$ and $B$ be real symmetric matrices with all eigenvalues strictly greater than $1$ . Let $\lambda$ be a real eigenvalue of matrix $AB$ . Prove that $|\lambda| > 1$ .

Official solution

Hidden so you can work on the problem first.

Proposed by Pavel Kozhevnikov, MIPT, Moscow.