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

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.