Question #40309

Let $A$ be a $7 \times 7$ invertible real matrix with $\det(A) = 2$, and let $B = \text{adj}(A)$ be its adjugate matrix. For any index sets $X, Y \subset \{1, 2, \dots, 7\}$, let $M_{X, Y}$ denote the submatrix of a matrix $M$ formed by taking the rows indexed by $X$ and columns indexed by $Y$, both in increasing order. Let $S = \{1, 2, 3, 4\}$ and $T = \{1, 2, 4, 6\}$. If $\det(A_{T^c, S^c}) = 5$ and $\det(A_{S^c, T^c}) = 7$, where $X^c$ denotes the complement of $X$ in $\{1, 2, \dots, 7\}$, what is the value of $\det(B_{S, T})$?