OIMU 2010 Problem 5

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