Linear Algebra – Determinant of a Special Symmetric Matrix

Question

If $A$ is a symmatric matrix of odd order with integer entries and the diagonal entries $0$
then $A$ has determinant value even.

I can prove the result if I can show that the eigenvalues of $A$ are integers,but I am unable to do that.
Thanks for any help.

Answer 1

As suggested by @Grigory M I post this as an answer :

A is congruent modulo 2 to a skew-symmetric matrix and an antisymmetric matrix of odd size has a zero determinant.