Question

If $\left|\begin{array}{ccc}1& x& x^2\\ x& x^2& 1\\ x^2& 1& x\end{array}\right|=3$, then find the value of $\left|\begin{array}{ccc}{x}^3-1& 0& x-x^4\\ 0& x-x^4& x^3-1\\ x-x^4& x^3-1& 0\end{array}\right|$

Answer 1

Let $D=|A|=\left|\begin{array}{ccc}1& x& x^2\\ x& x^2& 1\\ x^2& 1& x\end{array}\right|=3$
Determinant of co-factors
$D^c=\left|\begin{array}{ccc}{x}^3-1& 0& x-x^4\\ 0& x-x^4& x^3-1\\ x-x^4& x^3-1& 0\end{array}\right|$
$\Rightarrow D^c=|adjA|$ Since, $|A|=|A^T|$
$\Rightarrow D^c=|A{|}^2=9$
$\therefore \left|\begin{array}{ccc}{x}^3-1& 0& x-x^4\\ 0& x-x^4& x^3-1\\ x-x^4& x^3-1& 0\end{array}\right|=9$