Too large of a matrix

Since $\alpha$ , $\beta$ , and $\gamma$ are third roots of unity $1$ , $\omega$ , and $\omega^2$ and $\alpha$ is real, $\alpha = 1$ . Let $\beta = \omega$ and $\gamma = \omega^2$ .

$\begin{aligned} A & = \begin{vmatrix} \alpha & \alpha + \beta & \alpha + \beta + \gamma \\ 3\alpha & 4\alpha + 3\beta & 5\alpha + 4\beta + 3\gamma \\ 6\alpha & 9\alpha + 6\beta & 11\alpha + 9\beta + 6\gamma \end{vmatrix} \\ & = \begin{vmatrix} 1 & 1 + \omega & \color{#3D99F6} 1 + \omega + \omega^2 \\ 3 & 4 + 3\omega & \color{#3D99F6} 5 + 4\omega + 3\omega^2 \\ 6 & 9 + 6\omega & \color{#3D99F6} 11 + 9\omega + 6\omega^2 \end{vmatrix} & \small \color{#3D99F6} \text{Note that }1+\omega +\omega^2 = 0 \\ & = \begin{vmatrix} 1 & 1 + \omega & \color{#3D99F6} 0 \\ 3 & 4 + 3\omega & \color{#3D99F6} 2 + \omega \\ 6 & 9 + 6\omega & \color{#3D99F6} 5 + 3\omega \end{vmatrix} \\ & = \begin{vmatrix} 1 & 1 + \omega & 0 \\ 0 & 1 & 2 + \omega \\ 0 & 3 & 5 + 3\omega \end{vmatrix} \begin{matrix} \\ \small \color{#3D99F6} \text{row 2 - 3 row 1} \\ \small \color{#3D99F6} \text{row 3 - 6 row 1} \end{matrix} \\ & = \begin{vmatrix} 1 & 1 + \omega & 0 \\ 0 & 1 & 2 + \omega \\ 0 & 0 & -1 \end{vmatrix} \begin{matrix} \\ \\ \small \color{#3D99F6} \text{row 3 - 2 row 2} \end{matrix} \\ & = \boxed{-1} \end{aligned}$