Prove the following identities –

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 C₂→ C₂ – pC₁, we get

Applying C₃→ C₃ – qC₁, we get

Applying C₃→ C₃ – pC₂, we get

Applying C₂→ C₂ – C₁, we get

Applying C₃→ C₃ – C₁, we get

Expanding the determinant along R₁, we have

Δ = 1[(1)(7) – (3)(2)] – 0 + 0

∴ Δ = 7 – 6 = 1

Thus,