Exercise 25, § 8

Let $A$ be a commutative ring containing the field $\mathbf{Q}$ of rational numbers and let $X$ be a square matrix of even order $2n \geq 2$ over $A$. Let $L$ be the subset $\{1, 2, \ldots, n+1\}$ of $\{1, 2, \ldots, 2n\}$ and $L'$ its complement. Show the following formula ("Redei's identity"):

(*)

where $H$ runs through the set of subsets of $n$ elements of $\{1, 2, \ldots, 2n\}$ containing 1 and $K$ the set of subsets of $n$ elements of $L$ and $r = \operatorname{Card}(H \cap L')$. (Evaluate the coefficient of a term $x_{\sigma(1), 1} x_{\sigma(2), 2} \ldots x_{\sigma(2n), 2n}$ in the right-hand side of (*) for an arbitrary permutation $\sigma \in S_{2n}$, noting that this coefficient can only be $\neq 0$ if $H = \sigma(K)$; show that it can only be $\neq 0$ if $\sigma(L) = L$.)