linear algebra - Finite set of matrices closed under multiplication

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.

Answer

Pick any matrix $A$ in your finite set, and look at its powers.
Since they are in a finite set, they repeat at some point :
there exists $p,k \ge 1$ such that $A^p = A^{p+k}$.
In particular, if $m$ is a multiple of $k$ and greater than $p$, then $A^{2m}=A^m$.

Thus $A^m$ is the matrix of a projection. It shouldn't be too hard now to show that its trace is an integer between $0$ and $n$
And the trace is $0$ if and only if $A$ was nilpotent in the first place. Thus you would need to accept $0$ or require that the set contains non nilpotent matrices for the statement of the exercise to be correct.