28.2 The Binet Cauchy Formula
The Binet Cauchy formula is a generalization of the theorem which says the determinant
of a product is the product of the determinants. The situation is illustrated in the
Theorem 28.2.1 Let A be an n × m matrix with n ≥ m and let B be a m × n matrix.
Also let Ai
be the m × m submatrices of A which are obtained by deleting n − m rows and let Bi be
the m×m submatrices of B which are obtained by deleting corresponding n−m columns.
Proof: This follows from a computation. By Corollary 4.7.5 on Page 183,
Now denote by Ik
one subsets of
Thus there are C
of these. Then the above equals
since there are m
! ways of arranging the indices