Editing Cauchy–Binet formula (section)

== Generalization ==

The Cauchy–Binet formula can be extended in a straightforward way to a general formula for the [[minor (linear algebra)|minors]] of the product of two matrices. Context for the formula is given in the article on [[Minor (linear algebra)#Other applications|minors]], but the idea is that both the formula for ordinary [[matrix multiplication]] and the Cauchy–Binet formula for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices. 
Suppose that '''A''' is an ''m'' × ''n'' matrix, '''B''' is an ''n'' × ''p'' matrix, ''I'' is a [[subset]] of {1,...,''m''} with ''k'' elements and ''J'' is a subset of {1,...,''p''} with ''k'' elements. Then  
:<math>[\mathbf{AB}]_{I,J} = \sum_{K} [\mathbf{A}]_{I,K} [\mathbf{B}]_{K,J}\,</math>
where the sum extends over all subsets ''K'' of {1,...,''n''} with ''k'' elements.  Note the notation <math>[\mathbf{M}]_{I,J}</math> means the determinant of the matrix formed by taking only the rows of {{mvar|M}} with index in {{mvar|I}} and the columns with index in {{mvar|J}}.