---
tag: 08WE2W7DYS
kind: statement
role: theorem
name: Cauchy–Binet formula
msc: [15A15]
uses: [0C3CFHDN6R, 0SD528P0EK]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy–Binet_formula
    locator: Statement
same_as:
  wikidata: Q45286
  wikipedia: 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|0BT9BYWVRY]] 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.