Let

Given that Δ = 0.

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 R₂→ R₂ – R₁, we get

Applying R₃→ R₃ – R₁, we get

Expanding the determinant along C₃, we have

⇒ Δ = (c – z)[0 – (–x)(y)] – 0 + z[(a)(y) – (–x)(b – y)]

⇒ Δ = (c – z)(xy) + z[ay + xb – xy]

⇒ Δ = cxy – xyz + ayz + bxz – xyz

∴ Δ = ayz + bxz + cxy – 2xyz

We have Δ = 0

⇒ ayz + bxz + cxy – 2xyz = 0

⇒ ayz + bxz + cxy = 2xyz

Thus, when.