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Question

Evaluate $∣ ∣ ∣ ∣ \begin{matrix} 1 & x & y 1 & x + y & y 1 & x & x + y \end{matrix} ∣ ∣ ∣ ∣$

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Solution

Let $Δ = ∣ ∣ ∣ ∣ \begin{matrix} 1 & x & y 1 & x + y & y 1 & x & x + y \end{matrix} ∣ ∣ ∣ ∣$
Using $R_{1} = R_{1} - R_{2}, R_{2} = R_{2} - R_{3}$
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} 0 & - y & 0 0 & y & - x 1 & x & x + y \end{matrix} ∣ ∣ ∣ ∣$
Now expanding along first column we get,
$Δ = 1 [- x (- y) - 0] = x y$

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