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 – R₁, we get

Taking the term p common from R₂, we get

Applying C₁→ C₁ – C₂, we get

Expanding the determinant along C₁, we have

Δ = p(2 – x)[(1)(1) – (1)(x)]

∴ Δ = p(2 – x)(1 – x)

The given equation is Δ = 0.

⇒ p(2 – x)(1 – x) = 0

Assuming p ≠ 0, we get

⇒ (2 – x)(1 – x) = 0

Case – I:

2 – x = 0 ⇒ x = 2

Case – II:

1 – x = 0 ⇒ x = 1

Thus, 1 and 2 are the roots of the given determinant equation.