Let Δ=∣∣
∣∣xyx+yyx+yxx+yxy∣∣
∣∣
Using C1=C+C2,C2=C2+C3
Δ∣∣
∣
∣∣2(x+y)yx+y2(x+y)x+yx2(x+y)xy∣∣
∣
∣∣=2(x+y)∣∣
∣∣1yx+y1x+yx1xy∣∣
∣∣
Using R1−R2,R2=R2−R3
Δ=2(x+y)∣∣
∣∣0−xy0yx−y1xy∣∣
∣∣
Now expanding along first column we get,
Δ=2(x+y)[−x(x−y)−y2]=−2(x+y)(x2+y2−xy)