Evaluate


Let Δ =

Applying Elementary Transformations.


Applying R1 R1 + R2 + R3, we have


Δ =


Δ = 2 (x + y)


Applying C2 C2 – C1 and C3 C3 – C1, we have


Δ = 2 (x + y)


Expanding along R1, we have


Δ = 2 (x + y) [1 (x × (-x) – (-y) × (x – y)) – 0 + 0]


Δ = 2 (x + y) [-x2 + y (x – y)]


Δ = 2 (x + y) [-x2 + xy – y2]


Δ = -2 (x + y) [x2 – xy + y2]


Δ = -2 (x3 + y3)


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