Let A be a 33 real matrix.

Question

Let $A$ be a $3	imes3$ real matrix. If $\det\big(2\operatorname{Adj}(2\operatorname{Adj}(\operatorname{Adj}(2A)))\big)=2^{41}$, then the value of $\det(A^2)$ equals

Answer: 4

Accepted Answer

Using $\operatorname{adj}(kM)=k^{n-1}\operatorname{adj}M$ (with $n=3$) repeatedly:
$$2\operatorname{Adj}(2\operatorname{Adj}(\operatorname{Adj}(2A)))=2\operatorname{Adj}\big(2\cdot4\operatorname{Adj}(\operatorname{Adj}A)\cdot|2|^{\cdots}\big),$$
which simplifies (per the source's step-by-step expansion) to $2^{11}\cdot\operatorname{Adj}(\operatorname{Adj}(\operatorname{Adj}A))=2^{11}\cdot|A|^{3-2}\cdot(\cdots)$, ultimately giving
$$\det\big(2\operatorname{Adj}(2\operatorname{Adj}(\operatorname{Adj}(2A)))\big)=2^{33}|A|^8=2^{41}\Rightarrow|A|^8=2^8\Rightarrow|A|=\pm2.$$

So $\det(A^2)=|A|^2=4$.

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