Prove that the following determinant equals $0$

Question

We have a $n\times n$ matrix $A=(a_{i,j})$.
If $i=j$, then $a_{i,j}=1-n$. otherwise, $a_{i,j}=1$. Show that $|A|=0$.
I tried using gauss elimination but it just gets too complicated. I also tried to do $R_i \rightarrow R_i-R_1/(1-n)$ for all $i>1$, and then expand over the first column, but it also didn't work for me. Can someone please help?

Answer 1

Hint:$$ A\left(\begin{array}{c}1\\1\\\vdots\\1\end{array}\right)=0. $$