Show that : `|x y z x^2y^2z^2x^3y^3z^3|=x y z(x-y)(y-z)(z-x)dot`

Answer

Proved

Explanation

To prove
$$\left|\begin{array}{ccc}x & y & z \\ x^{2} & y^{2} & z^{2} \\ x^{3} & y^{3} & z^{3}\end{array}\right|=x y z(x-y)(y-z)(z-x) .$$ LHS $$=\left|\begin{array}{ccc}x & y & z \\ x^{2} & y^{2} & z^{2} \\ x^{3} & y^{3} & z^{3}\end{array}\right|=x y z\left|\begin{array}{ccc}1 & 1 & 1 \\ x & y & z \\ x^{2} & y^{2} & z^{2}\end{array}\right|$$
[taking `x, y` and `z` common from `C_{1}, C_{2}` and `C_{3}`, respectively]


On applying `C_{1} \rightarrow C_{1}-C_{2}` and then
`C_{2} \rightarrow C_{2}-C_{3}` we get
$$
\mathrm{LHS}=x y z\left|\begin{array}{ccc}
0 & 0 & 1 \\
...