Determinants: A Special Condition Under what conditions is
$$ \det(A_1 + \cdots + A_n) = \det(A_1)+\cdots+\det(A_n), $$
just curious.
 A: I'm not sure if you're looking for necessary and sufficient conditions, or just a list of sufficient ones.  Here is one sufficient condition: find any solution to $x_1+x_2+\ldots+x_n=0$, say $(a_1,a_2,\ldots,a_n)$, and let $A_i$ be a matrix full of $a_i$'s.  Then both sides are zero.
A bit more general: let $a_1+\ldots+a_{n-1}=0$, and define $A_i$ as above for $i=1,\ldots,n-1$.  Then for any $A_n$, both sides are equal to $\operatorname{det}(A_n).$
A: Of course we're only interested in $n \geq 2$. Let $A_{jk}$ be the $k$th row of $A_j$. The determinant map $\det(A_j)$ can be instead interpreted as an alternating $n$-multilinear map $\delta: \mathbb{F}^n \times \ldots \times \mathbb{F}^n \rightarrow \mathbb{F}$ such that $\delta(e_1,\ldots,e_n) = 1$; then $\det(A_j) = \delta(A_{j1}, \ldots, A_{jn})$. Then the condition
$$ \det(A_1 + \ldots + A_n)  = \det(A_1) + \ldots + \det(A_n)$$
can be rewritten as
$$ \delta(A_{11} + \ldots + A_{n1}, \ldots, A_{1n} + \ldots + A_{nn}) = \delta(A_{11}, \ldots, A_{1n}) + \ldots + \delta(A_{n1}, \ldots, A_{nn})$$
The LHS can be expanded, by multilinearity, and subtracting from the RHS, we obtain
$$ \sum_{i \in \{1, \ldots, n\}^n, i(k) \text{ not constant}} \delta(A_{i(1),1}, \ldots, A_{i(n),n}) = 0$$
In other words, the sum of all determinants of matrices where you mix and match all possible arrangements of rows (of the correct index) from any of the matrices $A_1, \ldots, A_n$, such that not all the rows come from the same matrix, must equal zero. The above condition is, of course, necessary and sufficient, but not probably not very satisfying since it is merely a bunch of algebraic manipulations. But I think that is all you can get with such a general algebraic condition.
You could construct a variety of sufficient conditions as special cases of the above condition, though again I'm not sure which would be particularly interesting without being unnecessarily restrictive or obvious. For example, you could require that each summand is zero; this would be equivalent to saying that every matrix where you mix up rows between the matrices $A_1, \ldots, A_n$ is not invertible.
