$\alpha, \beta, \gamma$ are the roots of given equation, Therefore $\alpha+\beta+\gamma=-\mathrm{a}$ $\alpha \beta+\beta \gamma+\gamma \alpha=0$ and $\alpha \beta \gamma=-\mathrm{b}$ Now, $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{array}\right|=-(\alpha+\beta+\gamma)$ $\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\alpha \beta-\beta \gamma-\gamma \alpha\right)$ $=-(\alpha+\beta+\gamma)\left[(\alpha+\beta+\gamma)^{2}\right.$ $\left.=-(-\mathrm{a})\left(\mathrm{a}^{2}-0\right)=\mathrm{a}^{3} \quad-3(\alpha \beta+\beta \gamma+\gamma \alpha)\right]$