Question

Solve: $\left|\begin{array}{ccc}x& x^2& yz\\ y& y^2& zx\\ z& z^2& xy\end{array}\right|=(x-y)(y-z)(z-x)(xy+yz+zx)$.

Answer 1

Now,

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

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

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

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

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

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

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

$=(y-x)(z-x)(z-y)\left\{x\right(x+y+z)-1(x^2-yz\left)\right\}$

$=(x-y)(y-z)(z-x)(xy+yz+zx)$