JEE Main 2025 Apr 7 Shift 1, Mathematics Q15: Algebra Of Matrices

$|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}A))|=81$

$\Rightarrow |\mathrm{adj}A{|}^4=81$

$\Rightarrow |\mathrm{adj}A|=3$

$\Rightarrow |A{|}^2=3$

$\Rightarrow |A|=\sqrt{3}$

$\left(|A|^4\right)^{\frac{(n-1)^2}{2}} = |A|^{3n^2-5n-4}$

$\Rightarrow 2(n-1{)}^2=3n^2-5n-4$

$\Rightarrow 2n^2-4n+2=3n^2-5n-4$

$\Rightarrow n^2-n-6=0$

$\Rightarrow (n-3)(n+2)=0$

$\Rightarrow n=3,-2$

$\sum _{n\in S}|A^{n^2+n}|$

$=|A^2|+|A^{12}|$

$=(\sqrt{3}{)}^2+(\sqrt{3}{)}^{12}$

$=3+729$

$\boxed{732}$