Question

Let $A$ be a $3 \times 3$ matrix such that $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A ))|=12^4$ Then $\mid A ^{-1}$ adj $A \mid$ is equal to

• A. 1
• B. $\sqrt{6}$
• C. 12
• D. $2 \sqrt{3}$

Answer 1

Answer: (D)

Given $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} . A ))|=12^4$ $\Rightarrow| A |^{( n -1)^3}=12^4$
Given $n =3$
$\Rightarrow| A |^8=12^4 $
$\Rightarrow| A |^2=12$
$ | A |=2 \sqrt{3}$
We are asked
$ \left| A ^{-1} \cdot \operatorname{adj} A \right|$
$ =\left| A ^{-1}\right| \cdot|\operatorname{adj} A | $
$ =\frac{1}{| A |} \cdot| A |^{3-1} $
$ =| A |=2 \sqrt{3}$