1. When does $\det(A+B)=\det(A)+\det(B)$ hold?
2. I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.



When does the determinant distribute over addition?
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 
$$x\_n=\frac{1-U}{U-V},\;\;U=\prod\_{i=1}^{n-1}(1+x\_i),\;\;V=\prod\_{i=1}^{n-1}x\_i$$


