Tricky determinant , I seem to be close to computing it

Question

Compute $\begin{vmatrix} 1+x_1 & x_2 & x_3 & ... & x_n \\ x_1 & 1+x_2 & x_3 & ... & x_n\\ . &.&.&&. \\ . &.&.&&. \\ . &.&.&&. \\ x_1 & x_2 & x_3 & ... & 1+x_n \\ \end{vmatrix}\\ $. I tried to subtract the kth row from the (k-1)th,but I can't work it out.

Answer 1

if you subtract off $I,$ you have a rank one matrix with eigenvalues $$ (0,0,0,\ldots,0, x_1+x_2+\cdots +x_n) $$ Add back the $I$ each eigenvalue increases by 1