Evaluate 1&x&y 1&x+y&y 1&x&x+y - Vidyadip

Question

Evaluate $\begin{vmatrix}1&\text{x}&\text{y}\\1&\text{x}+\text{y}&\text{y}\\1&\text{x}&\text{x+y}\end{vmatrix}$

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Answer

$\triangle=\begin{vmatrix}1&\text{x}&\text{y}\\1&\text{x}+\text{y}&\text{y}\\1&\text{x}&\text{x+y}\end{vmatrix}$
Applying $R_2 \rightarrow R_2 - R_1$ and $R_3 \rightarrow R_3 - R_1,$ we have$:$
$\triangle=\begin{vmatrix}1&\text{x}&\text{y}\\0&\text{y}&0\\0&0&\text{x}\end{vmatrix}$
Expanding along $C_1,$ we have$:$
$\triangle = 1(xy - 0) = xy$

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