Exercise 10, § 8

Let $\Gamma, \Delta$ be the determinants of two matrices $U, V$ of order $n$; let H and K be any two subsets of $\{1, n\}$ with $p$ elements and $(i_k)$ (resp. $(j_k)$) the sequence obtained by arranging the elements of H (resp. K) in increasing order; let $\Gamma_{H, K}$ be the determinant obtained by replacing in $U$ the column of index $i_k$ by the column of $V$ of index $j_k$, for $1 \leq k \leq p$; similarly let $\Delta_{K, H}$ be the determinant obtained by replacing in $V$ the column of index $j_k$ by the column of $U$ of index $i_k$, for $1 \leq k \leq p$. Show that, for every subset $H$ of $\{1, n\}$ with $p$ elements,

where $K$ runs through the set of subsets of $\{1, n\}$ with $p$ elements (use formulae (21) and (22) of no. 6).