Prove that determinant of matrix equal to n

Question

Prove that determinant of matrix $D_n$ (square $n$ x $n$ matrix) is equal to $n$. $$ \begin{matrix} 1 & -1 & -1 & \cdots & -1 \\ 1 & 1 & & & \\ 1 & & 1 & & \\ \vdots & & & \ddots & \\ 1 & & & & 1 \\ \end{matrix} $$

I tried to find some recurrence here when calculating using LaPlace expansion, but with no success.

Answer 1

Consider the first column $C_1$, and replace it by $C_1-\sum_{j=2}^nC_j$. This column becomes $\pmatrix{n\\0\\\vdots\\0}$.