Let A be a 3 × 3 real matrix. If det ( 2 Adj ( 2 Adj ( Adj ( 2 A ) ) ) ) = 2 41 , then the value of det…

$\therefore \left|KA\right|=K^n\left|A\right|,$ If $A$ is $n\times n$ matrix

$|Adj\mathrm{A}|=|\mathrm{A}{|}^{\mathrm{n}-1}$ If $A$ is $n\times n$ matrix

$|2Adj(2Adj(Adj(2A\left)\right)\left)\right|=2^{41}$

$\Rightarrow 2^3|Adj(2Adj(Adj(2 \mathrm{A}\left)\right)\left)\right|=2^{41}$
$\Rightarrow |Adj(2Adj(Adj(2 \mathrm{A}\left)\right)\left)\right|=2^{38}$
$\Rightarrow \left(\right|2Adj(Adj(2 \mathrm{A}\left)\right)|{)}^2=2^{38}$
$\Rightarrow |2Adj(Adj(2 \mathrm{A})\left)\right|=2^{19}$
$\Rightarrow 2^3|Adj(Adj(2 \mathrm{A})\left)\right|=2^{19}$
$\Rightarrow |Adj(Adj(2 \mathrm{A})\left)\right|=2^{16}$
$\Rightarrow \left(\right|Adj(2 \mathrm{A})|{)}^2=2^{16}$
$\Rightarrow |Adj(2 \mathrm{A}\left)\right|=2^8$
$\Rightarrow \left|\right(2 \mathrm{A}){|}^2=2^8$
$\Rightarrow \left|\right(2 \mathrm{A}\left)\right|=2^4$
$\Rightarrow 2^3| A|=2^4$
$\Rightarrow \left|A\right|=2$

So, $\left|A^2\right|={\left|A\right|}^2=4$