Steady Determinant!

Let $\omega = e^{2\pi i/3}$ and let $M = A + \omega B + \omega^2 C$ . Using $1 + \omega + \omega^2 = 0$ and $\overline{\omega} = \omega^2$ , we get:

$M\overline{M} = (A+\omega B + \omega^2 C)(A + \omega^2 B+ \omega C)$

$= A^2 + B^2 + C^2 + \omega^2(AB+BC+CA) + \omega(BA+CB+AC)$

$= (1+\omega^2)(AB+BC+CA) + \omega(BA+CB+AC)$

$= -\omega(AB+BC+CA - (BA+CB+AC) )$ .

Hence:

$\text{det}(M\overline{M}) = (-\omega)^n [\text{det}((AB-BA)+(BC-CB)+(CA-AC))]$

$\Rightarrow |\text{det}(M)|^2 = (-\omega)^n [\text{det}((AB-BA)+(BC-CB)+(CA-AC))]$

But $|\text{det}(M)|^2 \in \mathbb R$ . Therefore $(-\omega)^n [\text{det}((AB-BA)+(BC-CB)+(CA-AC))] \in \mathbb R$ .

Since $n$ is not a multiple of $3$ , therefore, the only option is $\text{det}((AB-BA)+(BC-CB)+(CA-AC)) = 0$ .