The following problem is from Golan's linear algebra book. I have been unable to make headway.
Problem: Let $n\in\mathbb{N}$ and $U$ be a non-empty finite subset of the $n\times n$ matrices over $\mathbb{C}$ which is closed under the multiplication of matrices and contains more than just the zero matrix. Show there exists a matrix $A$ in $U$ satisfying $tr(A)\in \{1,...,n\}$
EDIT: As noted in the comments, this problem is incorrect as stated. Perhaps it is correct if we allow $0$ to be in the set of desired values, or if we require the set to contain a non-singular matrix? Any help reformulating the problem would be much appreciated.