Exercise 1, § 8

In a matrix U of order n, if, for each index i, the column of index i is replaced by the sum of the columns of index $\neq i$, the determinant of the matrix obtained is equal to $(-1)^{n-1}(n-1) \det U$. If in U, for each i, the sum of the columns of index $\neq i$ is subtracted from the column of index i, the determinant obtained is equal to $-(n-2)2^{n-1} \det U$.