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.



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Expansion of $\det(A+B)$
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. 


