Question

Evaluate `|(x, y, x+y),(y, x+y, x),(x+y, x, y)|`

Answer 1

Applying C₁ → C₁ + C₂ + C₃, we get

Δ = `|(2(x + y),y,x+y),(2(x + y), x+y,x),(2(x + y),x,y)|`

Taking 2(x + y) common from C₁, we get

= `2(x + y)[(1,y,x+y),(1,x+y,x),(1,x,y)]`

Applying R₂ → R₂ − R₁ and R₃ → R₃ − R₁, we get

= `2(x + y)[(1,y,x+y),(0,x,-y),(0,x-y,-x)]`

Expanding along C₁, we get

= 2(x + y) [x(−x) − (−y) (x − y)]

= 2(x + y) [−x² + xy − y²]

= −2(x + y) (x² − xy + y²)

= −2(x³ + y³)