<s>[INST] Find the real number $x$ such that $\det A=0$ $$A= \begin{bmatrix} 1 & x & x^2 & x^3\\ x & x^2 & x^3 & 1 \\ x^2 & x^3 & 1 & x \\ x^3 & 1 & x&x^2  \end{bmatrix}$$
I don't know how to approach this. I'd like to figure this out step-by-step so any suggestions would be greatly appreciated. Thanks in advance!
 [/INST] By performing column operations we can evaluate the determinant to be $(x^4-1)^3$. So, the roots are $1$, $-1$, $i$ and $-i$.
Try $C_2 \to C_2 - xC_1$ and $C_4 \to C_4 - xC_3$. This will easily simplify the determinant into:
$\det A= \left| \begin{matrix} 1 & 0 & x^2 & 0\\ x & 0 & x^3 & 1 - x^4\\ x^2 & 0 & 1 & 0 \\ x^3 & 1-x^4 & x&  0\end{matrix}\right|$
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