Justify determinant using properties of determinants

Question

I'd like to understand how to prove that the determinant of the matrix below is zero, without actually calculating it, just based on the determinant properties

$$ \begin{vmatrix} x & y & z \\ z+y & z+x & x+y \\ 1 & 1 & 1 \\ \end{vmatrix} $$

I searched everywhere for but couldn't find a question like mine, so if it's duplicate, I apologize in advance

Answer 1

Adding row two to row one doesn't affect the determinant, so it equals $$\begin{vmatrix} x+y+z & x+y+z & x+y+z \\ z+y & z+x & x+y \\ 1 & 1 & 1 \\ \end{vmatrix}.$$ If one row is a multiple of another, the determinant is zero.