Question

Without expanding, prove that
$\Delta =\left|\begin{array}{ccc}x+y& y+z& z+x\\ z& x& y\\ 1& 1& 1\end{array}\right|=0$

Answer 1

Applying $R_1\to R_1+R_2$ to $\Delta$, we get
$\Delta =\left|\begin{array}{ccc}x+y+z& x+y+z& x+y+z\\ z& x& y\\ 1& 1& 1\end{array}\right|=0$
since the elements of $R_1$ and $R_3$ are proportional, $\Delta =0$