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If $A,B\in{\bf M}_n(k)$, then the following formula holds true: $$\det(A+B)=\sum_{r=0}^n\sum_{|I|=|J|=r}\epsilon(I,I^c)\epsilon(J,J^c)A\binom IJ B\binom{I^c}{J^c}.$$ In this formula, $I$ and $J$ are ordered (increasingly) $r$-uplets in $[1,n]$, $A\binom IJ$ is the corresponding minor, $I^c$ is the ordered complement of $I$ and $\epsilon(I,I^c)$ is the signature of the permutation thus defined.

I should like to have a reference for this formula. Thanks in advance.

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The formula appears in

Marvin Marcus "determinants of sums". College mathematics journal, March 1990.

https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/determinants-of-sums

The author notes that it is hard to track it's origins, but mentions previous references in the text, in particular to a book of himself in 1975.

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    $\begingroup$ Astonishing ! I keep amazed by the efficiency of MO. Thanks a lot. $\endgroup$ – Denis Serre Mar 18 at 13:24

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