If we want:

\[

\frac{1}{3}

\begin{bmatrix}

1 & 1 & 1 \\

1 & a & a^2 \\

1 & a^2 & a

\end{bmatrix}

=

\frac{1}{3}

\begin{bmatrix}

1 & 1 & 1 \\

1 & b^2 & b \\

1 & b & b^2

\end{bmatrix}

\]

And $\displaystyle a = e^{\frac{2\pi i}{3}}$, how much is $\displaystyle b$ to fulfill the condition?

So if I cancel $\displaystyle \frac{1}{3} $on both sides and try to get the inverse of one matrix I don't get $\displaystyle b = e^{\frac{-2\pi i}{3}}$ which is correct.

Thanks for the help.