Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Under what conditions does $\det(A+B)=\det(A)+\det(B)$ hold?
We all know that for any two positive semi-definite matrices $A$ and $B$, $\det(A+B) \geq \det(A)+\det(B)$ and strict inequality holds when $A$ and $B$ are both positive definite matrices having order $\geq 2$. My question is that under what conditions on $A$ (positive definite) and $B$ (positive semi-definite) the reverse also holds true, that is, the equality holds? Please help.