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Question Number 363 by rajabhay last updated on 25/Jan/15

If α, β and γ are roots of the equation x^3 +px+q=0, p≠0, q≠0 then find the value of the determinant. determinant ((α,β,γ),(β,γ,α),(γ,α,β))

Ifα,βandγarerootsoftheequationx3+px+q=0,p≠0,q≠0thenfindthevalueofthedeterminant.|αβγβγαγαβ|

Commented by 123456 last updated on 24/Dec/14

determinant ((α,β,γ),(β,γ,α),(γ,α,β))=3αβγ−(α^3 +β^3 +γ^3 ) x^3 +px+q=0 α+β+γ=0 αβ+αγ+βγ=p αβγ=−q

|αβγβγαγαβ|=3αβγ−(α3+β3+γ3)x3+px+q=0α+β+γ=0αβ+αγ+βγ=pαβγ=−q

Answered by prakash jain last updated on 24/Dec/14

determinant ((α,β,γ),(β,γ,α),(γ,α,β))= determinant (((α+β+γ),β,γ),((β+γ+α),γ,α),((γ+α+β),α,β)) (C_1 =C_1 +C_2 +C_3 ) = determinant ((0,β,γ),(0,γ,α),(0,α,β))=0

|αβγβγαγαβ|=|α+β+γβγβ+γ+αγαγ+α+βαβ|(C1=C1+C2+C3)=|0βγ0γα0αβ|=0

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