Let $n$ be a positive integer and let $U$ be a finite subset of $M_{n\times n}(\mathbb{C})$ which is closed under multiplication of matrices. Show that there exists a matrix $A$ in $U$ satisfying $\text{trace}(A) \in \{1,...,n\}.$
I was thinking if we look at this problem by contradiction. Then we have that for all matrices $A$ $\text{trace}(A)>n$ or $\text{trace}(A)=0.$ In the first case, we get that $\det(A)<1$ and the second case we get that $\det{A}=0.$ I am not sure how to proceed further, so any hint would be much appreciated.