\(\left|\begin{array}{ccc}1 & x & x^{2} \\ x^{2} & 1 & x \\ x & x^{2} & 1\end{array}\right| C_{1} \rightarrow C_{1}+C_{2}+C_{3}\)
\(=\left|\begin{array}{ccc} 1+x+x^{2} & x & x^{2} \\ 1+x+x^{2} & 1 & x \\ 1+x+x^{2} & x^{2} & 1 \end{array}\right|=\left(1+x+x^{2}\right)\left|\begin{array}{ccc} 1 & x & x^{2} \\ 1 & 1 & x \\ 1 & x^{2} & 1 \end{array}\right| \)
\(R_{1} \rightarrow R_{1}-R_{2}, R_{2} \rightarrow R_{2}-R_{3} \)
\(=\left(1+x+x^{2}\right)\left|\begin{array}{ccc} 0 & x-1 & x^{2}-x \\ 0 & 1-x^{2} & x-1 \\ 1 & x^{2} & 1 \end{array}\right|\)
\(=\left(1+x+x^{2}\right)\left|\begin{array}{ccc} 0 & -(1-x) & -x(1-x) \\ 0 & (1+x)(1-x) & -(1-x) \\ 1 & {x}^{2} & 1 \end{array}\right|\)
\(=(1-x)^2\left(1+x+x^2\right)\left|\begin{array}{ccc}
0 & -1 & -x \\
0 & 1+x & -1 \\
1 & x^2 & 1
\end{array}\right| \)
\(=(1-x)^2\left(1+x+x^2\right) \times 1\left|\begin{array}{cc}
-1 & -x \\
1+x & -1
\end{array}\right| \)
\( =(1-x)^2\left(1+x+x^2\right)(1+x(1+x))\)
\(=(1-x)^2\left(1+x+x^2\right)\left(1+x+x^2\right)\)
\(=(1-x)^2\left(1+x+x^2\right)^2=\left\{\left(1-x+x^2\right)\left(1+x+x^2\right)\right\}^2=\left(1-x^3\right)^2=\text { R.H.S. }
\)
