Question

The value of the determinant ${\displaystyle \left|\begin{array}{ccc}1& \text{x}& {\text{x}}^3\\ 1& \text{y}& {\text{y}}^3\\ 1& \text{z}& {\text{z}}^3\end{array}\right|}$ is ____________.

पर्याय

• 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 ${\displaystyle \left|\begin{array}{ccc}1& \text{x}& {\text{x}}^3\\ 1& \text{y}& {\text{y}}^3\\ 1& \text{z}& {\text{z}}^3\end{array}\right|}$ is (x – y)(y – z)(z – x)(x + y + z).

Explanation:

${\displaystyle \left|\begin{array}{ccc}1& \text{x}& {\text{x}}^3\\ 1& \text{y}& {\text{y}}^3\\ 1& \text{z}& {\text{z}}^3\end{array}\right|}$

Apply, R₁→ R₁ - R₂, R₂ → R₂ - R₃ 

${\displaystyle \left|\begin{array}{ccc}0& \text{x - y}& {\text{x}}^3-{\text{y}}^3\\ 0& \text{y - z}& {\text{y}}^3-{\text{z}}^3\\ 1& \text{z}& {\text{z}}^3\end{array}\right|}$

${\displaystyle \Rightarrow \left(\text{x - y}\right)\left(\text{y - z}\right)\left|\begin{array}{ccc}0& 1& {\text{x}}^2+{\text{y}}^2+\text{xy}\\ 0& 1& {\text{y}}^2+{\text{z}}^2+\text{yz}\\ 1& \text{z}& {\text{z}}^3\end{array}\right|}$

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

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