Skew Product of Symmetric Matrices

Question

Let $A,B$ be two real $n\times n$ symmetric matrices. Is it true that $AB=-BA$ implies $AB=0$? Note that this condition is equivalent to $AB=-(AB)^T-B^TA^T=-BA$, i.e. it is equivalent to $AB$ being skew-symmetric. Thus the question is really saying: is it true that the product of two symmetric matrices is skew-symmetric if and only if the product is 0?

Answer 1

$$\pmatrix{1&1\cr1&-1\cr}\pmatrix{1&-1\cr-1&-1\cr}=\pmatrix{0&-2\cr2&0\cr}$$