By using properties of determinant, prove the following : |(x+λ,2x,2x)(2x,x+λ,2x)(2x,2x,x+λ) = (5x + λ)(λ - x)^2 ​

Question

By using properties of determinant, prove the following :

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

Answer 1

LHS,

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

Expanding along C₁, we get

= (5x + λ) (λ - x)² 

= RHS.