TITLE: When does the determinant distribute over addition?
QUESTION [7 upvotes]: When does $\det(A+B)=\det(A)+\det(B)$ hold?
I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.

REPLY [15 votes]: let me assume $A$ is invertible, then you ask when
$$\det(1+X)=1+\det X,\;\;X=A^{-1}B $$
so if $X$ has eigenvalues $x_i$, $i=1,2,\ldots n$, you would need
$$\prod_{i}(1+x_i)=1+\prod_i x_i$$
basically you can take arbitrary values for $x_1,x_2,\ldots x_{n-1}$ and then the only requirement is that 
$$x_n=\frac{1-U}{U-V},\;\;U=\prod_{i=1}^{n-1}(1+x_i),\;\;V=\prod_{i=1}^{n-1}x_i$$