Question

If D = `[(0, aα^2, aβ^2),(bα + c, 0, aγ^2),(bβ + c, (bγ + c), 0)]` is a skew-symmetric matrix (where α, β, γ are distinct) and the value of `|((a + 1)^2, (1 - a), (2 - c)),((3 + c), (b + 2)^2, (b + 1)^2),((3 - b)^2, b^2, (c + 3))|` is λ then the value of |10λ| is ______.

पर्याय

• 300

• 400

• 500

• 600

Answer 1

If D = `[(0, aα^2, aβ^2),(bα + c, 0, aγ^2),(bβ + c, (bγ + c), 0)]` is a skew-symmetric matrix (where α, β, γ are distinct) and the value of `|((a + 1)^2, (1 - a), (2 - c)),((3 + c), (b + 2)^2, (b + 1)^2),((3 - b)^2, b^2, (c + 3))|` is λ then the value of |10λ| is 600.

Explanation:

From skew-symmetric matrix,

aα² + bα + c = 0,

aβ² + bβ + c = 0

aγ² + bγ + c = 0

⇒ ax² + bx + c = 0 has three roots.

a = b = c = 0

λ = `|(1, 1, 2),(3, 4, 1),(9, 0, 3)|` = –60

⇒ |10λ| = |10 × –60| = 600