Question:
Evaluate $\left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ x & x+y\end{array}\right|$
Solution:
$\Delta=\left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$
Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1}$, we have:
$\Delta=\left|\begin{array}{lll}1 & x & y \\ 0 & y & 0 \\ 0 & 0 & x\end{array}\right|$
Expanding along $\mathrm{C}_{1}$, we have:
$\Delta=1(x y-0)=x y$
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