Too large of a matrix

Algebra Level 3

\[\begin{vmatrix} \alpha & \alpha +\beta & \alpha +\beta +\gamma \\ 3\alpha & 4\alpha +3\beta & 5\alpha +4\beta +3\gamma \\ 6\alpha & 9\alpha +6\beta & 11\alpha +9\beta +6\gamma \end{vmatrix}\]

Given the constants \( \alpha, \beta, \gamma \) are roots to the equation \(z^3 = 1\) with \(\alpha \) being a real number. What is the value of the determinant above?

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