Determinant of a sum of matrices I would like to know if the following formula is well known and get some references for it.
I don't know yet how to prove it (and I am working on it), but I am quite sure of its validity, after having performed a few symbolic computations with Maple.
Given $n$ square matrices $A_1,\ldots,A_n$ of size $m<n$ :

$$\sum_{p=1}^n(-1)^p\sum_{1\leqslant i_1<\cdots<i_p\leqslant n}\det(A_{i_1}+\cdots+A_{i_p})=0$$

For example, if $A,B,C$ are three $2\times2$ matrices, then :
$$\det(A+B+C)-\left[\det(A+B)+\det(A+C)+\det(B+C)\right]+\det(A)+\det(B)+\det(C)=0$$

HINT:
The determinant of an $n\times n$ matrix is a form of degree $n$. Forms come from multilinear forms. 
Consider $M$ an abelian group. For $a \in M$, denote by $a^{[n]}$ the element $a\otimes a \otimes \ldots \otimes a\in M^{\otimes n}$. Let now $a_i\in M$, $i \in I$, finitely many elements in $M$. Let's try to find 
$$\sum_{J\subset I}(-1)^{|I|-|J|}(\sum_{i \in J} a_i)^{[n]}$$
Consider a product $a_{i_1}\otimes \ldots \otimes a_{i_n}$. It appears in the above sum with the coefficient
$$\sum_{J\subset K \subset I}(-1)^{|I| - |J|}$$ where $J=\{i_1, \ldots, i_n \}$. This is $0$ for $J\ne I$ and $1$ for $J=I$. ( a Möbius function)
Therefore
$$\sum_{J\subset I}(-1)^{|I|-|J|}(\sum_{i \in J} a_i)^{[n]}=\sum_{\phi\colon \{1,\ldots n\}\to I,\phi\  \text{surjective}}a_{\phi(1)}\otimes \ldots a_{\phi(n)}$$
Particular cases:


*

*$|I|>n$, we get $0$, the result desired. 

*$|I|=n$, we get $\sum_{\phi\colon \{1,\ldots n\}\to I,\phi\  \text{bijective}}a_{\phi(1)}\otimes \ldots a_{\phi(n)}$
