TITLE: Find out trace of a given matrix $A$ with entries from $\mathbb{Z}_{227}$
QUESTION [9 upvotes]: Let $A$ be a $227\times 227$ matrix with entries in $\mathbb{Z}_{227}$ such that all of its eigen values are distinct. What would be its trace?
I think it is zero by adding all 227 elements but i am not sure.
Edited: Here I have assumed that eigenvalues are in a base field.

REPLY [5 votes]: The trace of a matrix is defined to be the sum of its diagonal elements. The trace is equal to the sum of the eigenvalues, i.e. the roots of the characteristic polynomial, when the characteristic polynomial splits over the underlying field $F$. Consequently, we can not say anything in the present case without further information. 
For example, if all eigenvalues are in $\mathbb{Z}_{227}$ and they are distinct, then yes, the trace is equal to zero. But in general, the eigenvalues will not be all of them inside $\mathbb{Z}_{227}$.