Determinant with Cofactor or Expansion Relationship

The question requires relating a determinant to minors, cofactors, or sub-determinants (e.g., expanding a 3×3 determinant to find a 2×2 determinant).

jee-main 2024 Q69 View
Let $\alpha \beta \neq 0$ and $A = \left[ \begin{array} { r r r } \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ - \beta & \alpha & 2 \alpha \end{array} \right]$. If $B = \left[ \begin{array} { r r r } 3 \alpha & - 9 & 3 \alpha \\ - \alpha & 7 & - 2 \alpha \\ - 2 \alpha & 5 & - 2 \beta \end{array} \right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname { det } ( A B )$ is equal to :
(1) 64
(2) 216
(3) 343
(4) 125