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Assume $A,B \in \mathbb{R}^{n \times n}$ and let $M$ denote the set of $2^n$ matrices we get by replacing, in turn, each subset of columns of $A$ with the corresponding columns of $B$. Then $$\det(A+B) = \sum_{C \in M}\det(C).$$

I found this result on a previous answer, but neither was it named nor was any proof provided. Since I can't find the name or proof anywhere, I ask the question here. What is this formula called? What is its proof?

Rodrigo de Azevedo
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squingo44
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    Hint : The determinant is multilinear w.r.t. the columns. – TheSilverDoe Aug 26 '21 at 13:29
  • Do you agree with my edits? I assumed it was a quote. Please let me know if I assumed incorrectly. – Rodrigo de Azevedo Aug 26 '21 at 13:37
  • Of course, there is a similar identity where one instead replaces some rows of $A$ with the corresponding rows of $B$. Also, one could instead replace some columns (or rows) of $B$ with the corresponding columns (or rows) of $A$. – Geoffrey Trang Aug 26 '21 at 14:18

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