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 R₂→ R₂ – R₃, we get

Applying C₂→ C₂ – C₁, we get

Applying C₃→ C₃ – C₁, we get

Expanding the determinant along R₂, we have

Δ = – (1)[(–4 – x)(3) – (6)(x – 8)]

⇒ Δ = – [–12 – 3x – 6x + 48]

⇒ Δ = – [– 9x + 36]

∴ Δ = 9x – 36

The given equation is Δ = 0.

⇒ 9x – 36 = 0

⇒ 9x = 36

∴ x = 4

Thus, 4 is the root of the given determinant equation.