Question

If true Enter ${}^{\prime }{1}^{\prime }$ else ${}^{\prime }{0}^{\prime }.$

$\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ x+y& x& y\end{array}\right|=-2(x+y)(x^2+y^2-xy)$

Answer 1

Let $\Delta =\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ x+y& x& y\end{array}\right|$

Using $C_1=C+C_2,C_2=C_2+C_3$

$\Delta =\left|\begin{array}{ccc}2(x+y)& y& x+y\\ 2(x+y)& x+y& x\\ 2(x+y)& x& y\end{array}\right|=2(x+y)\left|\begin{array}{ccc}1& y& x+y\\ 1& x+y& x\\ 1& x& y\end{array}\right|$

Using $R_1-R_2,R_2=R_2-R_3$

$\Delta =2(x+y)\left|\begin{array}{ccc}0& -x& y\\ 0& y& x-y\\ 1& x& y\end{array}\right|$

Now expanding along first column we get,
$\Delta =2(x+y)[-x(x-y)-y^2]=-2(x+y)(x^2+y^2-xy)$