Cauchy–Binet formula

Statement

Let $K$ be a commutative ring, let $m \le n$, let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix over $K$. For a subset $S \subseteq \{1, \dots, n\}$ with $m$ elements, let $A_S$ be the $m \times m$ matrix of the columns of $A$ in $S$. Let $B^S$ be the $m \times m$ matrix of the rows of $B$ in $S$. Then $$\det(AB) = \sum_{S} \det(A_S) \det(B^S),$$ where the sum runs over all subsets $S$ of $\{1, \dots, n\}$ with $m$ elements.