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.


The formula appears in

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


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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