Question

Evaluate the following determinant without expanding:

`|(5, 5, 5),(a, b, c),(b + c, c + a, a + b)|`

Answer 1

Δ = `|(5, 5, 5),(a, b, c),(b + c, c + a, a + b)|`

R₃ → R₃ + R₂

Δ = `|(5, 5, 5),(a, b, c),(a + b + c, a + b + c, a + b + c)|`

= `5 xx (a + b + c) xx |(1, 1, 1),(a, b, c),(1, 1, 1)|`

= 5(a + b + c) × 0    ...(As R₁ and R₃ are identical)

= 0