Question

Let $A$ and $B$ be two $3\times 3$ matrices such that $AB=I$ and $|A|=\frac{1}{8}$ then $|\mathrm{adj}(\mathrm{Badj}(2A))|$ is equal to

• A. 16
• B. 32
• C. 64
• D. 128

Answer 1

Answer: (C)

$AB=i$
$\mid \mathrm{adj}({B\mathrm{adj}(2A)|=|B\mathrm{adj}(2A)|}^2$
$=|B{|}^2|\mathrm{adj}(2A){|}^2$
$=|B{|}^2{\left(|2A{|}^2\right)}^2=|B{|}^2{\left(2^6|A{|}^2\right)}^2$
$|A|=\frac{1}{8}\text{ and }|AB|=1\Rightarrow |A||B|=1$
$\Rightarrow \frac{1}{8}|B|=1$
$\Rightarrow |B|=8$
$\text{ required value }=64$