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
Best Answer
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.