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Question
Find the value of the determinant given below, without expanding it at any stage.
`|(βγ, 1, α(β + γ)),(γα, 1, β(γ + α)),(αβ, 1, γ(α + β))|`
Solution
Given: `|(βγ, 1, α(β + γ)),(γα, 1, β(γ + α)),(αβ, 1, γ(α + β))|`
= `|(βγ, 1, αβ + αγ),(γα, 1, βγ + βα),(αβ, 1, γα + γβ)|`
Applying C3 → C1 + C3
= `|(βγ, 1, αβ + βγ + αγ),(γα, 1, βγ + βα + γα),(αβ, 1, αβ + γβ + γα)|`
= `(αβ + βγ + γα)|(βγ, 1, 1),(γα, 1, 1),(αβ, 1, 1)|`
= (αβ + βγ + γα) × 0 ...(∵ C2 and C3 are similar)
= 0
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