Let

Taking 2 common from C₂, we get

Recall that the value of a determinant remains same if we apply the operation R_i→ R_i + kR_j or C_i→ C_i + kC_j.

Applying R₂→ R₂ – R₁, we get

Applying R₃→ R₃ – R₁, we get

We have the identity a³ – b³ = (a – b)(a² + ab + b²)

Taking (b – a) and (c – a) common from R₂ and R₃, we get

We know that the sign of a determinant changes if any two rows or columns are interchanged.

By interchanging C₁ and C₂, we get

Expanding the determinant along C₁, we have

Δ = –2(b – a)(c – a)(1)[(b² + ba + a²) – (c² + ca + a²)]

⇒ Δ = 2(a – b)(c – a)[b² + ba + a² – c² – ca – a²]

⇒ Δ = 2(a – b)(c – a)[b² + ba – c² – ca]

⇒ Δ = 2(a – b)(c – a)[b² – c² + (ba – ca)]

⇒ Δ = 2(a – b)(c – a)[(b – c)(b + c) + (b – c)a]

⇒ Δ = 2(a – b)(c – a)(b – c)(b + c + a)

∴ Δ = 2(a – b)(b – c)(c – a)(a + b + c)

Thus,