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In linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with entries from any commutative ring. == Statement == Let ''A'' be an ''m''×''n'' matrix and ''B'' an ''n''×''m'' matrix. Write () for the set , and for the set of ''m''-combinations of () (i.e., subsets of size ''m''; there are of them). For , write ''A''(),''S'' for the ''m''×''m'' matrix whose columns are the columns of ''A'' at indices from ''S'', and ''B''''S'',() for the ''m''×''m'' matrix whose rows are the rows of ''B'' at indices from ''S''. The Cauchy–Binet formula then states : Example: taking ''m'' = 2 and ''n'' = 3, and matrices and , the Cauchy–Binet formula gives the determinant: : Indeed , and its determinant is −28, which is also the value given by the right hand side of the formula. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cauchy–Binet formula」の詳細全文を読む スポンサード リンク
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