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.

Remarks

For $m = n$, the sum has one term, and the formula is $\det(AB) = \det A \det B$. If $m > n$, then $\det(AB) = 0$. For $m = 1$, it is the formula for the product of a row and a column. The determinants $\det(A_S)$ and $\det(B^S)$ are minors of order $m$.

Explanations

L1 human

The Cauchy–Binet formula computes the determinant of a product of two rectangular matrices. It sums products of matching minors of the two factors. It generalizes the product rule for determinants of square matrices.