Question

$\left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ x^3& y^3& z^3\end{array}\right|=xyz(x-y)(y-z)(z-x)$

• A. True
• B. False

Answer 1

The correct option is A True

Let $A=\left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ x^3& y^3& z^3\end{array}\right|$

Taking $x,y,z$ common from $C_1,C_2,C_3$ respectively, we get,

$A=xyz\left|\begin{array}{ccc}1& 1& 1\\ x& y& z\\ x^2& y^2& z^2\end{array}\right|$

Applying $C_2\to C_2-C_1$ and $C_3\to C_3-C_1$, we get,

$A=xyz\left|\begin{array}{ccc}1& 0& 0\\ x& y-x& z-x\\ x^2& y^2-x^2& z^2-x^2\end{array}\right|$

Taking $(y-x)$ and $(z-x)$ common from $C_2$ and from $C_3$

$A=xyz(y-x)(z-x)\left|\begin{array}{ccc}1& 0& 0\\ x& 1& 1\\ x^2& y+x& z+x\end{array}\right|$

Expanding along $R_1$,

$A=xyz(y-x)(z-x)(z+x-y-x)$

$A=xyz(x-y)(y-z)(z-x)$