Trace of product of semidefinite matrices is nonnegative

I want to prove this:

$$A$$ is a symmetric positive semi-definite matrix $$\Leftrightarrow$$ $$tr(AB) \geq 0$$ $$\forall$$ B positive semi-definite.

I tried using eigenvalues, because they all have to be non-negative, but that didn't help me too much, as the eigenvalues of $$AB$$ are different from those of $$A$$ or $$B$$.

I would like some tips about where to start.

3) If a matrix $$A$$ is symmetric positive-semidefinite, so is $$M^TAM$$ for any $$M$$.