$\begin{vmatrix} 1 & x & x^2 \\ x^2 & 1 & x \\ x & x^2 & 1 \end{vmatrix}$ Applying $R_1$ → $R_1 + R_2 + R_3$, we have: Δ = = 1 + x + $x^2$ $\begin{vmatrix} 1 & 1 & 1 \\ x^2 & 1 & x \\ x & x^2 & 1 \end{vmatrix}$ Applying $C_2$ → $C_2 - C_1$ and $C_3$ → $C_3 - C_1$, we have: Δ = 1 + x + $x^2$ $\begin{vmatrix} 1 & 0 & 0 \\ x^2 & 1-x^2 & x-x^2 \\ x & x^2-x & 1-x \end{vmatrix}$ = (1 + x + $x^2$) (1 - x) (1 - x) $\begin{vmatrix} 1 & 0 & 0 \\ x^2 & 1+x & x \\ x & -x & 1 \end{vmatrix}$ = (1 - $x^3$) (1 - x) $\begin{vmatrix} 1 & 0 & 0 \\ x^2 & 1+x & x \\ x & -x & 1 \end{vmatrix}$ Expanding along $R_1$, we have: Δ = (1 - $x^3$) (1 - x) (1) $\begin{vmatrix} 1+x & x \\ -x & 1 \end{vmatrix}$ = (1 - $x^3$) (1 - x) (1 + x + $x^2$) = (1 - $x^3$) (1 - $x^2$) = $(1 - x^3)^2$ Hence proved.