Let Δ = Applying Elementary Transformations. Applying R 1 → R 1 + R 2 + R 3 , we have Δ = Δ = 2 (x + y) Applying C 2 → C 2 – C 1 and C 3 → C 3 – C 1 , we have Δ = 2 (x + y) Expanding along R 1 , we have Δ = 2 (x + y) [1 (x × (-x) – (-y) × (x – y)) – 0 + 0] Δ = 2 (x + y) [-x 2 + y (x – y)] Δ = 2 (x + y) [-x 2 + xy – y 2 ] Δ = -2 (x + y) [x 2 – xy + y 2 ] Δ = -2 (x 3 + y 3 )

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)