Question

Let $A$ be a $3\times 3$ matrix such that $|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}A))|=12^4$ 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 $|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}.A))|=12^4$ $\Rightarrow |A{|}^{(n-1{)}^3}=12^4$
Given $n=3$
$\Rightarrow |A{|}^8=12^4$
$\Rightarrow |A{|}^2=12$
$|A|=2\sqrt{3}$
We are asked
$\left|A^{-1}\cdot \mathrm{adj}A\right|$
$=\left|A^{-1}\right|\cdot |\mathrm{adj}A|$
$=\frac{1}{|A|}\cdot |A{|}^{3-1}$
$=|A|=2\sqrt{3}$