Question

Find the determinant:

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

Answer 1

Now,

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

[$R_2^{\prime }=R_2-R_1$ and $R_3^{\prime }=R_3-R_1$]

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

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

$=(y-x)(z-x)\left\{\right(z+x)-(y+x\left)\right\}$

$=(y-x)(z-x)\left\{\right(z-y\left)\right\}$

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