Question

Prove 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

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

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

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

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

RHS:-${(1-x^3)}^2$

$\therefore$LHS=RHS

For al values of x