Prove the following using properties of determinants:

Let Δ =

Applying C₁ → C₁ + C₂ + C₃, we get

Δ =

Taking out common from C₁, we get

Δ =

Applying R₂→ R₂ – R₁ and R₃→ R₃ – R₁, we get

Δ =

Expanding along C₁, we get

Δ = 2(a+b+c)[1.{(a+b+c)²-0}]

Δ = 2(a+b+c)³ = RHS

Hence proved.