Binet-Cauchy Identity

Theorem

 * $\displaystyle \left({\sum_{i=1}^n a_i c_i}\right) \left({\sum_{j=1}^n b_j d_j}\right) = \left({\sum_{i=1}^n a_i d_i}\right) \left({\sum_{j=1}^n b_j c_j}\right) + \sum_{1 \le i < j \le n} \left({a_i b_j - a_j b_i}\right) \left({c_i d_j - c_j d_i}\right)$

where all of the $a, b, c, d$ are elements of a commutative ring.

Thus the identity holds for $\Z, \Q, \R, \C$.

Proof
Expanding the last term:

Hence the result.

Note
This is in fact a special case of the Cauchy-Binet Formula.

It is also known as Binet's formula.