Prove the following identities –

(CBSE 2011)

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 (5x + 4) 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 = (5x + 4)[(1)(4 – x)(4 – x)]

∴ Δ = (5x + 4)(4 – x)²

Thus,