Question

Evaluate $\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ x+y& x& y\end{array}\right|$

Answer 1

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

$=\left|\begin{array}{ccc}2(x+y)& y& x+y\\ 2(x+y)& x+y& x\\ 2(x+y)& x& y\end{array}\right|$
(using $C_1\to C_1+C_2+C_3$)
$=2(x+y)\left|\begin{array}{ccc}1& y& x+y\\ 1& x+y& x\\ 1& x& y\end{array}\right|$ |Taking out 2(x+y) common from $C_1$|
$=2(x+y)\left|\begin{array}{ccc}1& y& x+y\\ 0& x& -y\\ 0& x-y& -x\end{array}\right|$
(using $R_2\to R_2-R_1;R_3\to R_3-R_1$)
Expanding along $R_1$, we get
$=2(x+y)\times 1(-x^2+y(x-y\left)\right)=2(x+y)(-x^2+xy-y^2)=-2(x+y)(x^2-xy+y^2)=-2(x^3+y^3)$