$\underbrace{\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \ldots . .(\mathrm{a})))}_{2024 \text { times }}=|\mathrm{A}|^{(\mathrm{n}-1)^{2024}} $

$ \quad=|\mathrm{A}|^{2024} $

$ =2^{2^{2024}}$

$2^{2024}=\left(2^2\right) 2^{2022}=4(8)^{674}=4(9-1)^{674}$

$\Rightarrow 2^{2024} \equiv 4(\bmod 9) $

$\Rightarrow 2^{2024} \equiv 9 \mathrm{~m}+4, \mathrm{~m} \leftarrow \text { even } $

$2^{9 \mathrm{~m}+4} \equiv 16 \cdot\left(2^3\right)^{3 \mathrm{~m}} \equiv 16(\bmod 9) $

$\quad \equiv 7$