On the difference of two positive semi-definite matrices

On the difference of two positive semi-definite matrices

I am relatively new to linear algebra, and have been struggling with a
problem for a few days now. Say we have two positive semi-definite
matrices A and B, and further assume that A and B are such that A - B is
also positive semi-definite. Can it be shown that det(A) >= det(B)? In my
own attempts, I can see that Tr(A) >= Tr(B), but I do not think this is
enough to prove the desired result. Perhaps there is something to be said
about the relative magnitudes of the eigenvalues of A and B, but I can't
see it. In any case, I would appreciate any help. Thanks a lot.