Prove that

Let Δ =

Applying Row Transformations

R₂→ R₂ – R₁

Δ =

R₃→ R₃ – R₁

Δ =

Taking (β – α)(γ – α) from R₂ and R₃ respectively

Δ = (β – α) (γ – α)

Applying R₃→ R₃ – R₂, we have

Δ = (β – α) (γ – α)

Expanding along R₃, we have

Δ = (β – α) (γ – α) [0 (α² × (-1) – (β + γ) × (β + γ) – (γ – β)((-1) × α – 1 × (β + γ) + 0 (α × (β + γ) – 1 × α²)

Δ = (β – α) (γ – α) [0 – (γ – β)( - α - β – γ) + 0]

Δ = (β – α) (γ – α) (γ – β) (α + β + γ)

Δ = (α – β) (β – γ) (γ – α) (α + β + γ)