JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Determinants

For a $3\times 3$ matrix, $|\mathrm{adj}(\mathrm{adj}(M))|=|M{|}^{(n-1{)}^2}=|M{|}^4$, so $|\mathrm{adj}(\mathrm{adj}(2A))|=|2A{|}^4=(2^3|A|{)}^4=2^{12}|A{|}^4$. Taking $5!$ from the first row, $6!$ from the second and $7!$ from the third cancels the factor $\frac{1}{5!6!7!}$ and leaves $|A|=\left|\begin{array}{ccc}1& 6& 42\\ 1& 7& 56\\ 1& 8& 72\end{array}\right|$. The operations $R_2\to R_2-R_1$ and $R_3\to R_3-R_2$ reduce this to $\left|\begin{array}{ccc}1& 6& 42\\ 0& 1& 14\\ 0& 1& 16\end{array}\right|=2$, so $|A|=2$ and $|\mathrm{adj}(\mathrm{adj}(2A))|=2^{12}\cdot 2^4=2^{16}$. The required value is $2^{16}$.