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.