Question

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

Answer 1

Let $A=\left|\begin{array}{ccc}1& x& y\\ 1& x+y& y\\ 1& x& x+y\end{array}\right|=\left|\begin{array}{ccc}1& x& y\\ 0& y& 0\\ 0& 0& x\end{array}\right|$
(using $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$)
Expanding along $C_1$, we get $A=1(yx-0)=xy$