Exercise 8, § 8

Let $\Delta = \det(\alpha_{ij})$ be the determinant of a matrix $U$ of order $n$ and $\sigma$ a permutation belonging to $\mathfrak{S}_n$; for each index $i$ ($1 \leq i \leq n$) let $\Delta_i$ be the determinant obtained by replacing the element $\alpha_{ij}$ in $U$ by $\alpha_{i, \sigma(j)}$ for $1 \leq j \leq n$; if $p$ is the number of indices invariant under the permutation $\sigma$, show that the sum of the $n$ determinants $\Delta_i$ ($1 \leq i \leq n$) is equal to $p\Delta$ (same method as in Exercise 7).