Let $A$ be a $3 imes 3$ real matrix. If $\operatorname{det}(2 \operatorname{Adj}(2 \operatorname{A

Question

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

Professor PC Thomas Classes · Answer

(4) : Given, $\operatorname{det}(2 \operatorname{Adj}(2 \operatorname{Adj}(\operatorname{Adj}(2 A))))$ $=2^{3} 	imes \operatorname{det}(\operatorname{Adj}(2 \operatorname{Adj}(\operatorname{Adj}(2 A))))$\ $=2^{3} 	imes \operatorname{det}[(2 \operatorname{Adj}(\operatorname{Adj}(2 A)))]^{2}$\ $=2^{9} 	imes(\operatorname{det}(\operatorname{Adj}(2 A)))^{4}=2^{9} 	imes(\operatorname{det}(2 A))^{8}$\ $=2^{9} 	imes\left(2^{3} \operatorname{det}(A)ight)^{8}=2^{41}$\ (Given)\ $\Rightarrow \quad 2^{33}(\operatorname{det} A)^{8}=2^{41}$\ $\Rightarrow \quad(\operatorname{det} A)^{8}=2^{8}$\ $\Rightarrow \quad \operatorname{det} A= \pm 2$\ $\Rightarrow \quad \operatorname{det}\left(A^{2}ight)=(\operatorname{det} A)^{2}=( \pm 2)^{2}=4$\