Exercise 7, § 8

Let $\Delta = \det(\alpha_{ij})$ be the determinant of a matrix $U$ of order $n$; for each index $i$ ($1 \leq i \leq n$) let $\Delta_i$ denote the determinant obtained by multiplying the element $\alpha_{ij}$ in $U$ by $\beta_j$ for $1 \leq j \leq n$. Show that the sum of the $n$ determinants $\Delta_i$ ($1 \leq i \leq n$) is equal to $(\beta_1 + \beta_2 + \cdots + \beta_n)\Delta$ (expand $\Delta_i$ by the row of index $i$).