In my textbook I have this question:
If $\omega = e^{\frac{2}{3}\pi i}$, and if $x + y + z = a$ , $x + \omega y + \omega^2 z = b$ and $x + \omega^2y + \omega z =c$, express $a+b+c$, $a + \omega^2 b + \omega c$ and $a + \omega b +\omega^2 c$ in terms of $x$, $y$ and $z$.
Hence solve for $x$, $y$ and $z$ the simultaneous equations
$$ \begin{cases} x + y +z =1 \\ x +\omega y + \omega^2 z = 2 \\ x + \omega^2 y +\omega z = 3 \end{cases} $$ Giving your answers in the form $p+q\omega$ where $p$ and $q$ are real numbers.
I was able to do the first part getting : $$ \begin{align} &a + b + c = 3x \\ &a + \omega^2 b + \omega c = 3y \\ &a + \omega b +\omega^2 c = 3z\end{align}$$
However I'm not sure where to start with the simultaneous equations.
If$$A=\begin{bmatrix}1&1&1\\1&\omega^2&\omega\\1&\omega&\omega^2\end{bmatrix},$$then your system is$$A.\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix}.$$Therefore, you can solve it doing$$\begin{bmatrix}x\\y\\z\end{bmatrix}=A^{-1}.\begin{bmatrix}1\\2\\3\end{bmatrix}.$$