Question

The value of `|(−a, ab, ac),(ab, −b^2, bc),(ac, bc, −c^2)|` is ______.

Options

• 0

• abc

• 4a²b²c²

• None of these

Answer 1

The value of `|(−a, ab, ac),(ab, −b^2, bc),(ac, bc, −c^2)|` is 4a²b²c².

Explanation:

Let A = `|(−a, ab, ac),(ab, −b^2, bc),(ac, bc, −c^2)|`

Taking a, b, c common from R₁, R₂ and R3 respectively, we get.

Δ = abc `|(−a, b, c),(a, −b, c),(a, b, −c)|` = a²b²c² `|(−1, 1, 1),(1, −1, 1),(1,1,−1)|`

[ taking a, b, c common from C₁, C2, C3 respectively]

= a²b²c² `|(−1, 0, 0),(1, 0, 2),(1,2,0)|`

(applying C₂ → C₂ + C₁, C₃→ C₃ + C₁)

= a²b²c².(-1) `|(0, 2), (2, 0)|` = a²b²c² (− 1) (0 − 4)

⇒ A = 4 a²b²c²