Prove that matrix expression is equal to $0$

Question

Suppose that $A$ and $X$ are square matrices and that I have the expression $$A^\top X + XA=0$$ and I know that $A\neq 0$ and $X^\top = X$. Does this mean that $X=0$? Or can the equality hold even when $X\neq 0$? If $X$ and $A$ were scalars the solution would be obvious but since they are matrices I am getting a bit confused.

Answer 1

Counter example

If $A^\top=-A$ and $X=I$ then $A^\top X + XA=0$

Answer 2

Consider $X=I$ and $A$ such that $A^T=-A.$ Then

$$A^TX+XA=-AI+IA=-A+A=0.$$