Cauchy-Binet formula

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 the entries from any commutative ring.

Let A be an m×n matrix and B an n×m matrix. Write [n] for the set { 1, ..., n }, and ${\displaystyle {\tbinom {[n]}{m}}}$ for the set of m-combinations of [n] (i.e., subsets of size m; there are ${\displaystyle {\tbinom {n}{m}}}$ of them). For ${\displaystyle S\in {\tbinom {[n]}{m}}}$, write A_[m],S for the m×m matrix whose columns are the columns of A at indices from S, and B_S,[m] for the m×m matrix whose rows are the rows of B at indices from S. The Cauchy–Binet formula then states

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