trace of a matrix in $\mathbb{Z}_{227}$

Question

Let $A$ be a $227 \times 227$ matrix with entries from $\mathbb{Z}_{227}$. If A has distinct eigenvalues what is the trace of $A$?

I am guessing the answer is zero. Since the eigenvalues of $A$ need not be in $\mathbb{Z}_{227}$, I dont know how to prove this.

Give me some hint to prove this. Thanks in advance.