Question

Using properties of determinants, prove that |(α, α², β + γ), (β, β², γ + α), (γ, γ², α + β)| = (β - γ)(γ - α)(α - β)(α + β + γ)

Answer 1

(using C₃ → C₃ + C₁)

(Taking out (α + β + γ) common from C₁)

(Using R₂ → R₂ – R₁ and R₃ → R₃ – R₁) 

Expanding along C₃, we get 

= (α + β + γ) [(β − α)(γ² −α²) −(γ −α)(β² − α²)] 

= (α + β + γ) [(β − α)(γ −α)(γ + α) −(γ − α)(β − α)(β + α)] 

= (α + β + γ) (β − α)(γ −α)[γ + α −β − α] 

= (α + β + γ) (β − α)(γ −α)(γ −β) 

= (α + β + γ) (α − β)(β − γ)(γ −α) = RHS