The value of the determinant xxyyzz|1xx31yy31zz3| is ____________. - Mathematics

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MCQ
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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)

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Solution

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)

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