Let A be a 3 × 3 real matrix.
If $\operatorname{det}(2 \operatorname{Adj}(2 \operatorname{Adj}(\operatorname{Adj}(2 \mathrm{~A}))))=2^{41}$, then the value
of $\operatorname{det}\left(\mathrm{A}^{2}\right)$ equal_________
$\operatorname{adj}(2 \mathrm{~A})=2^{2} \operatorname{adj} \mathrm{A}$
$\Rightarrow \operatorname{adj}(\operatorname{adj}(2 \mathrm{~A}))=\operatorname{adj}(4 \operatorname{adj} \mathrm{A})=16 \operatorname{adj}(\operatorname{adj} \mathrm{A})$
$=16|\mathrm{~A}| \mathrm{A}$
$\Rightarrow \operatorname{adj}(32|\mathrm{~A}| \mathrm{A})=(32|\mathrm{~A}|)^{2} \operatorname{adj} \mathrm{A}$
$12(32 \mid \mathrm{Al})^{2} \operatorname{ladj} \mathrm{Al}=2^{3}(32 \mid \mathrm{Al})^{6} \mid \operatorname{adj} \mathrm{Al}$
$2^{3} \cdot 2^{30}|\mathrm{~A}|^{6} \cdot|\mathrm{A}|^{2}=2^{41}$
$|\mathrm{A}|^{8}=2^{8} \Rightarrow|\mathrm{A}|=\pm 2$
$|\mathrm{A}|^{2}=|\mathrm{A}|^{2}=4$
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