Exercise 13, § 11

Let $\Gamma$ and $\Delta$ be two determinants of invertible matrices of order $n$ over a commutative ring and $\Gamma_{ij}$ the determinant obtained by replacing in $\Gamma$ the column of index $i$ by the column of $\Delta$ of index $j$. Show that

(Expand $\Gamma_{ij}$ by the $i$-th column and use Exercise 9.)