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

To prove: $\Delta =\left|\begin{array}{ccc}x+y& y+z& z+x\\ z& x& y\\ 1& 1& 1\end{array}\right|=0$
$\left|\begin{array}{ccc}x+y& y+z& z+x\\ z& x& y\\ 1& 1& 1\end{array}\right|$
Applying $R_1\to R_1+R_2$
$=\left|\begin{array}{ccc}x+y+z& x+y+z& x+y+z\\ z& x& y\\ 1& 1& 1\end{array}\right|$
Taking $(x+y+z)$ common from $R_1$
$=(x+y+z)\left|\begin{array}{ccc}1& 1& 1\\ z& x& y\\ 1& 1& 1\end{array}\right|$
$=0\{\because R_1\text{and}{R}_3\text{are identical}\}$