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Question
The value of the determinant `|(x , x + y, x + 2y),(x + 2y, x, x + y),(x + y, x + 2y, x)|` is ______.
Options
9x2(x + y)
9y2(x + y)
3y2(x + y)
7x2(x + y)
Solution
The value of the determinant `|(x , x + y, x + 2y),(x + 2y, x, x + y),(x + y, x + 2y, x)|` is 9y2(x + y).
Explanation:
Δ = `|(x , x + y, x + 2y),(x + 2y, x, x + y),(x + y, x + 2y, x)|`
[Applying C1 → C1 + C2 + C3]
= `|(3(x + y), x + y, x + 2y),(3(x + y), x, x + y),(3(x + y), x + 2y, x)|`
= `3(x + y)|(1, x + y, x + 2y),(1, x, x + y),(1, x + 2y, x)|`
[Applying R1 → R1 – R2 and R3 → R3 – R2]
= `3(x + y)|(0, y, y),(1, x, x + y),(0, 2y, -y)|`
= 3(x + y)[1(y(–y) – 2y(y)]
= 9y2(x + y)
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