Let α β ≠ 0 and A=[ array rrr β & α & 3 α & α & β -β & α & 2 α...

Equating co-factor fo $\mathrm{A}_{21}$$\begin{aligned} & \left(2 \alpha^2-3 \alpha\right)=\alpha \\ & \alpha=0,2 \text{ (accept) } \end{aligned}$Now, $2 \alpha^2-\alpha \beta=3 \alpha\begin{aligned} & \alpha=2 \quad \beta=1 \\ & |\mathrm{AB}|=|\mathrm{A} \operatorname{cof}(\mathrm{A})|=|\mathrm{A}|^3 \\ & \mathrm{ A}=\left|\begin{array}{lll} 1 & 2 & 3 \\ 2 & 2 & 1 \\ -1 & 2 & 4 \end{array}\right|=6-2(9)+3(6)=6 \end{aligned}$$