Using properties of determinants prove that: |(x+4, 2x,2x)(2x, x +4, 2x)(2x, 2x,x + 4)| = (5x + 4)(x - 4)^2

Question

Using properties of determinants prove that:

\(\begin{vmatrix} {\text{x}} +4 & 2{\text{x}} & 2{\text{x}} \\[0.3em] 2{\text{x}} & {\text{x}} +4 &2{\text{x}} \\[0.3em] 2{\text{x}} & 2{\text{x}} & {\text{x}} +4 \end{vmatrix}\) = (5x + 4)(x-4)²

|(x+4, 2x,2x)(2x, x +4, 2x)(2x, 2x,x + 4)| = (5x + 4)(x - 4)²

Answer 1

\(\begin{vmatrix} {\text{x}} +4 & 2{\text{x}} & 2{\text{x}} \\[0.3em] 2{\text{x}} & {\text{x}} +4 &2{\text{x}} \\[0.3em] 2{\text{x}} & 2{\text{x}} & {\text{x}} +4 \end{vmatrix}\)

= (5x + 4)(x - 4)[ - (x + 4) - 2x + 2x - 0 + 0 - ( - 2x)] 

[expansion by first row] 

= (5x + 4)(x - 4)( - x - 4 + 2x) 

=(5x + 4)(x - 4)²