Question

The value of the determinant `abs((1,"x","x"^3),(1,"y","y"^3),(1,"z","z"^3))` is ____________.

Options

• 2(x – y)(y – z)(z – x)

• (x – y)(y – z)(z – x)

• None of these

• (x – y)(y – z)(z – x)(x + y + z)

Answer 1

The value of the determinant `abs((1,"x","x"^3),(1,"y","y"^3),(1,"z","z"^3))` is (x – y)(y – z)(z – x)(x + y + z).

Explanation:

`abs((1,"x","x"^3),(1,"y","y"^3),(1,"z","z"^3))`

Apply, R₁→ R₁ - R₂, R₂ → R₂ - R₃ 

`abs((0, "x - y", "x"^3 - "y"^3),(0, "y - z", "y"^3 - "z"^3),(1,"z","z"^3))`

`=> ("x - y")("y - z") abs((0,1,"x"^2 + "y"^2 + "xy"),(0,1,"y"^2 + "z"^2 + "yz"),(1,"z","z"^3))`

= (x - y)(y - z)(y² + z² + yz - x² - y² - xy)

= (x - y)(y - z)(z - x)(x + y + z)