This is a True/False question. and the answer is given False.
A is symmetric and hence we can diagonalize A. say $DAD^{-1}$ gives our diagonal matrix $diag(a_1,a_2...a_r)$ , then by taking n-th power we easily get the $tr(A^n)= a_1^n+a_2^n+...a_r^n$ Now to show the claim is false I should play with $n$ right? In particular, if I put $n=1$, this gives no contradiction because there always exists such a symmetric matrix. I don't know where to go from here.