Question

Let $A$ be a $3 \times 3$ invertible matrix. If $\mid$ adj $(24 A) \mid= \operatorname{adj}(3 \operatorname{adj}(2 A )) \mid$, then $| A |^{2}$ is equal to :

• A. $6^{6}$
• B. $2^{12}$
• C. $2^{6}$
• D. $1$

Answer 1

Answer: (C)

$|\operatorname{adj}(24 A )|=|\operatorname{adj} 3(\operatorname{adj} 2 A )|$
$\Rightarrow|24 a |^{2}=(3 \operatorname{adj}(2 A ))^{2}$
$\Rightarrow\left(24^{3}| A |\right)^{2}=\left(3^{3}|\operatorname{adj}(2 A )|\right)^{2}$
$=3^{6}\left(|2 A |^{2}\right)^{2}$
$\Rightarrow 24^{6}| A |^{2}=\left(24^{3}| A |\right)^{2}=3^{6} \times 2^{12}| A |^{4}$
$\Rightarrow| A |^{2}=\frac{24^{6}}{3^{6} \times 2^{12}}=64$