Solve the following determinant equations:

Let

We need to find the roots of 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 C₁→ C₁ + C₂, we get

Applying C₁→ C₁ + C₃, we get

Taking the term (x + a + b + c) 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 = (x + a + b + c)(1)[(x)(x) – (0)(0)]

⇒ Δ = (x + a + b + c)(x)(x)

∴ Δ = x²(x + a + b + c)

The given equation is Δ = 0.

⇒ x²(x + a + b + c) = 0

Case – I:

x² = 0 ⇒ x = 0

Case – II:

x + a + b + c = 0 ⇒ x = –(a + b + c)

Thus, 0 and –(a + b + c) are the roots of the given determinant equation.