If a, b and c are all non-zero and , then prove that .

Let

Given that Δ = 0.

We can write the determinant Δ as

Taking a, b and c common from C₁, C₂ and C₃, we get

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 common from C₁, we get

Applying R₂→ R₂ – R₁, we get

Applying R₃→ R₃ – R₁, we get

Expanding the determinant along C₁, we have

We have Δ = 0.

It is given that a, b and c are all non-zero.

Thus, when and a, b, c are all non-zero.