Question

Evaluate|x y x+y y x+y x x+y x y|

Answer 1

Δ=|x y x+y y x+y x x+y x y|
Applying R1→R1+R2+R3,we have
Δ=|2(x+y) 2(x+y) 2(x+y) y x+y x x+y x y|
=2(x+y)|111  y x+y x x+y x y|
 Applying C2→C2-C1 and C3→C3-C1,we have
Δ=2(x+y)|100 y x x-y x+y -y -x|

Expanding along R1, we have:
Δ=2(x+y)[-x2+y(x-y)]
=-2(x+y)(x2+y2-yx)
=-2(x3+y3)