Prove the following identities –

(CBSE 2014)

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 C₁→ C₁ + C₂, we get

Applying C₁→ C₁ + C₃, we get

Taking the term (a + x + y + z) common from C₁, we get

Applying R₂→ R₂ – R₁, we get

Applying R₃→ R₃ – R₁, we get

Expanding the determinant along C₁, we have

Δ = (a + x + y + z)(1)[(a)(a) – (0)(0)]

⇒ Δ = (a + x + y + z)(a)(a)

∴ Δ = a²(a + x + y + z)

Thus,