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)

Answer 1

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)