Question

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

Answer 1

We have,

$\left|\begin{array}{ccc}x+y& y+z& z+x\\ z& x& y\\ 1& 1& 1\end{array}\right|$.

Using operation,

$R_1\to R_1+R_2$

$\left|\begin{array}{ccc}x+y+z& x+y+z& x+y+z\\ z& x& y\\ 1& 1& 1\end{array}\right|$.

$x+y+z$$\left|\begin{array}{ccc}1& 1& 1\\ z& x& y\\ 1& 1& 1\end{array}\right|$.

$(x+y+z)\times 0$=$0$

Hence, this is the answer.