Question

Using properties of determinants, show that $\left|\begin{array}{ccc}1& x& x^2\\ x^2& 1& x\\ x& x^2& 1\end{array}\right|={(1-x^3)}^2$.

Answer 1

$\left|\begin{array}{ccc}1& x& x^2\\ x^2& 1& x\\ x& x^2& 1\end{array}\right|=1(1-x^3)-x(x^2-x^2)+x^2(x^4-x)$

$=1-x^3+x^6-x^3$

$=x^6-{2x}^3+1$

$={\left({1-x}^3\right)}^2$

Hence proved.