Question

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

• A. 16
• B. 32
• C. 64
• D. 128

Answer 1

Answer: (C)

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