Is the following statement true: If $A\succeq 0$ and $B \preceq 0$ are two $n\times n$ real-valued matrices, then $AB \preceq 0$.

If not, is it true that $\forall x\ge 0$ (i.e., all vectors in the positive orthant), $x^TABx \le 0$?

If not, is the second statement true if all entries of $B$ are non-positive.

I'd really appreciate help with this.

Thanks!