Using properties of determinants, prove the following: |\table \alpha,\beta,\gamma;\alpha^2,\beta^2,\gamma^2;\beta+\gamma,\gamma+\alpha,\alpha+\beta| = (α - β) (β - γ) (γ - α) (α + β + γ)

CBSE Class 12 Math 2008 Solved Paper - Question 23

Question : 23 of 29

Marks: +1, -0

Using properties of determinants, prove the following:

|\table α,β,γ;α^2,β^2,γ^2;β+γ,γ+α,α+β|

= (α - β) (β - γ) (γ - α) (α + β + γ)

Solution:

Δ =

|\table α,β,γ;α^2,β^2,γ^2;β+γ,γ+α,α+β|

Applying

R_3

→

R_3+R_1

Δ =

|\table α,β,γ;α^2,β^2,γ^2;α+β+γ,α+β+γ,α+β+γ|

= α + β + γ

|\table α,β,γ;α^2,β^2,γ^2;1,1,1|

Applying

C_1 → C_1 - C_2

and

C_2→C_2-C_3

Δ = α + β + γ

|\table α-β,β-γ,γ;α^2-β^2,β^2-γ^2,γ^2;0,0,1|

= α + β + γ (α - β) (β - γ)

|\table 1,1,γ;αβ,β+γ,γ^2;0,0,1|

= α+ β + γ (α - β) (β - γ) [1 (β + γ) - 1 (α + β)]
= (α - β) (β - γ) (α + β + γ) (+ γ - α - β)
= (α - β) (β - γ) (γ - α) (α + β + γ)
Hence proved.

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