Prove the following identities –|(x+λ,2x,2x)(2x,x+λ,2x)(2x,2x,x+λ)| = (5x + λ)(λ - x)^2 ​

Question

Prove the following identities – 

\(\begin{vmatrix} x+\lambda &2x & 2x \\[0.3em] 2x & x+\lambda & 2x \\[0.3em] 2x &2x & x+\lambda \end{vmatrix}\) = (5x + λ)(λ - x)²

Answer 1

Let Δ = \(\begin{vmatrix} x+\lambda &2x & 2x \\[0.3em] 2x & x+\lambda & 2x \\[0.3em] 2x &2x & x+\lambda \end{vmatrix}\) 

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 R₁→ R₁ + R₂, we get 

Expanding the determinant along R₁, we have

Δ = (5x + λ)[(1)(λ – x)(λ – x)]

∴ Δ = (5x + λ)(λ – x)²

Thus,

\(\begin{vmatrix} x+\lambda &2x & 2x \\[0.3em] 2x & x+\lambda & 2x \\[0.3em] 2x &2x & x+\lambda \end{vmatrix}\) = (5x + λ)(λ - x)²