Question

Prove that:
$\left|\begin{array}{ccc}1& 1& 1\\ x& y& z\\ x^2& y^2& z^2\end{array}\right|=\left[(x-y)(y-z)(z-x)\right]$

Answer 1

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

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

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

taking out $(x-y)$ and $y-z$ from $C_1$ and $C_2$ respectively.

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

Expanding along $R_1$

$=(x-y)(y-z)\left[\right(y+z)-(x+y\left)\right]$

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