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 (3x – 2) 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 = (3x – 2)(1)[(3x – 11)(3x – 11) – (0)(0)]

⇒ Δ = (3x – 2)(3x – 11)(3x – 11)

∴ Δ = (3x – 2)(3x – 11)²

The given equation is Δ = 0.

⇒ (3x – 2)(3x – 11)² = 0

Case – I:

3x – 2 = 0

⇒ 3x = 2

Case – II:

(3x – 11)² = 0

⇒ 3x – 11 = 0

⇒ 3x = 11

Thus, and are the roots of the given determinant equation.