Question

Without expanding at any stage, find the value of the determinant:

`Δ = |(20, a, b + c),(20, b, a + c),(20, c, a + b)|`

Answer 1

Given:

`Δ = |(20, a, b + c),(20, b, a + c),(20, c, a + b)|`

Taking 20 common from C₁,

⇒ `Δ = 20 |(1, a, b + c),(1, b, a + c),(1, c, a + b)|`

Applying C₂ → C₂ + C₃,

`Δ = 20 |(1, a + b + c, b + c),(1, a + b + c, a + c),(1, a + b + c, a + b)|`

Taking (a + b + c) common from C₂,

`Δ = 20 (a + b + c) |(1, 1, b + c),(1, 1, a + c),(1, 1, a + b)|`

∵ C₁ = C₂

∴ Δ = 0