[Math] When is $\det(A+B)=\det(A)+\det(B)$ for positive definite $A$ and positive semidefinite $B$

Question

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.

Best regards,

Prasenjit Ghosh.

Answer 1

When $A\succ0$ and $B\succeq0$, $\det(A+B)=\det(A)+\det(B)$ if and only if $B=0$ or $n=1$. Hints:

1. $\det(A+B)=\det(A)+\det(B)$ is equivalent to $\det(I+A^{-1/2}BA^{-1/2})=1+\det(A^{-1/2}BA^{-1/2})$.
2. So you are essentially asking when does $\det(I+X)=1+\det(X)$ hold for a positive semidefinite $X$. WLOG you may assume that $X$ is diagonal.