Matrix determinant lemma

Question

I know that if $A,B$ are two positive semidefinite matrices of size $(N \times N)$ we have that:

$$\det(A+B) \geq \det(A) + \det(B)$$

My question is can anything similar be said, with $A$ and $B$ as defined above, for the relationship between $\det(A-B)$ and $\det(A)$ and $\det(B)$. For example, could the below be true and why?

$$\det(A-B) \leq \det(A) - \det(B)$$

Answer 1

Here is a counterexample ; with the symmetric positive definite matrices :

$$A=\begin{pmatrix}10&3\\3&1\end{pmatrix}, \ \ B=\begin{pmatrix}13&7\\7&5\end{pmatrix},$$

we have :

$$\det(A-B) > \det(A)-\det(B)$$