Prove the following identities –

Let

Multiplying a, b and c to R₁, R₂ and R₃, we get

Dividing C₁, C₂ and C₃ with a, b and c, 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 R₁→ R₁ + R₂, we get

Applying R₁→ R₁ + R₃, we get

Taking the term (a – b – c) common from R₁, we get

Applying C₂→ C₂ – C₁, we get

Applying C₃→ C₃ – C₁, we get

Expanding the determinant along R₁, we have

Δ = (ab + bc + ca)(1)[(ab + bc + ca)(ab + bc + ca)]

∴ Δ = (ab + bc + ca)³

Thus,