Prove the following identities –

Question

Philoid · Accepted Answer

Let

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 (3 + a) 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 = (3 + a)(1)[(a)(a) – 0]

⇒ Δ = (3 + a)(a²)

∴ Δ = a³ + 3a²

Thus,