Using properties of determinants prove that: |(x + λ, 2x,2x)(2x, x + λ, 2x)(2x,2x, x+λ)| = (5x + λ)(λ - x)^2

Question

Using properties of determinants prove that:

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

|(x + λ, 2x,2x)(2x, x + λ, 2x)(2x,2x, x+λ)| = (5x + λ)(λ - x)²

Answer 1

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

= (5x + )(x - λ)[ - (x + λ) - 2x + 2x - 0 + 0 - ( - 2x)] 

[expansion by first row] 

= (5x + )(x - λ)( - x - + 2x) 

= (5x + )(x - λ)²