Suppose *A* is an *m*-by-*n* matrix and *B* is an *n*-by-*m* matrix. If *S* is a subset of {1,...,*n*} with *m* elements, we write *A*_{S} for the *m*-by-*m* matrix whose columns are those columns of *A* that have indices from *S*. Similarly, we write *B*_{S} for the *m*-by-*m* matrix whose *rows* are those rows of *B* that have indices from *S*. The Cauchy-Binet formula then states

If *m* = *n*, i.e. if *A* and *B* are square matrices of the same format, then there is only a single admissible set *S*, and the Cauchy-Binet formula reduces to the ordinary multiplicativity of the determinant. If *m* = 1 then there are *n* admissible sets *S* and the formula reduces to that for the dot product. If *m* > *n*, then there is no admissible set *S* and the determinant det(*AB*) is zero (see empty sum).

The formula is valid for matrices with entries from any commutative ring. For the proof one writes the columns of *AB* as linear combinations of the columns of *A* with coefficients from *B*, uses the multilinearity of the determinant, and collects the terms that belong to a single det(*A*_{S}) together by exploiting the anti-symmetry of the determinant. The coefficient of det(*A*_{S}) is seen to be det(*B*_{S}) using the Leibniz formula for the determinant. This proof does not use the multiplicativity of the determinant; rather, the proof establishes it.

If *A* is a real *m*-by-*n* matrix, then det(*A* *A*^{T}) is equal to the square of the *m*-dimensional volume of the parallelepiped spanned in **R**^{n} by the *m* rows of *A*. Binet's formula states that this is equal to the sum of the squares of the volumes that arise if the parallelepiped is orthogonally projected onto the *m*-dimensional coordinate planes (of which there are C(*n*,*m*)). The case *m* = 1 of this statement talks about the length of a line segment: it is nothing but the Pythagorean theorem.

The Cauchy-Binet formula can be extended in a straight-forward way to a general formula for the minors of the product of two matrices. That formula is given in the article on minors.