Question

Prove the following :
$\left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right|=4xyz$

Answer 1

Apply $C_1-(C_2+C_3)$ to make one zero we get
$△=\left|\begin{array}{ccc}0& z& y\\ -2x& z+x& x\\ -2x& x& x+y\end{array}\right|$
Take out $-2x$ from $C_1$ and then apply $R_2-R_3$
$\Delta =-2x\left|\begin{array}{ccc}0& z& y\\ 0& z& -y\\ 1& x& x+y\end{array}\right|$
$=-2x(-zy-zy)=4xyz$.