Let A be a $3\times 3$ matrix and $\left|A \right|= 2$. If $\text{n}=\det \left(\underset{2024-\text{ times }}{\underbrace{\mathrm{adj}\left(\mathrm{adj}\right(\dots \dots \left(\mathrm{adj}A\right)}}\right)\left)\right)$ Then the remainder when n is divided by 9 is equal to_______.

|A| = 2
$\underset{2024\text{ times }}{\underbrace{\mathrm{adj}\left(\mathrm{adj}\right(\mathrm{adj}\dots \mathrm{..}\left(A\right)\left)\right)}}=|\text{A}{|}^{{(\text{n}-1)}^{2024}}=|\text{A}{|}^{2024}=2^{2024}$ 
$2^{2024}=\left(2^2\right)2^{2022}=4{\left(8\right)}^{674}=4{(9-1)}^{674}$ 
$\Rightarrow 2^{2024}\equiv 4\left(\mathrm{mod}9\right)\Rightarrow 2^{2024}\equiv 9 \text{m}+4, \text{m}\leftarrow \text{ even}$ 
$2^{9 \text{m}+4}\equiv 16·{\left(2^3\right)}^{3 \text{m}}\equiv 16\left(\mathrm{mod}9\right)\equiv 7$