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Roots of identity matrix with the same trace

Source: XIII Ibero American Olympiad For University Students

July 22, 2011
linear algebramatrixalgebrapolynomialinequalitieslinear algebra unsolved

Problem Statement

Let A,BA,B be matrices of dimension 2010×20102010\times2010 which commute and have real entries, such that A2010=B2010=IA^{2010}=B^{2010}=I, where II is the identity matrix. Prove that if tr⁡(AB)=2010\operatorname{tr}(AB)=2010, then tr⁡(A)=tr⁡(B)\operatorname{tr}(A)=\operatorname{tr}(B).
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