The value of $|adj(2A)$, if $A=[a_{ij}]_{3×3}$ and $|A|=2$ is :
Answer & explanation
Correct answer: option 1
Given: $A$ is a $3\times 3$ matrix, $|A|=2$.
For an $n\times n$ matrix, $\;|\text{adj}(A)|=|A|^{\,n-1}$.
Here $n=3\;\Rightarrow\;|\text{adj}(A)|=|A|^{2}=2^{2}=4$.
Now, $\text{adj}(kA)=k^{\,n-1}\,\text{adj}(A)$ for scalar $k$.
So, $\text{adj}(2A)=2^{3-1}\,\text{adj}(A)=2^{2}\,\text{adj}(A)=4\,\text{adj}(A)$.
Therefore, $|\text{adj}(2A)|=|4\,\text{adj}(A)|$.
$\text{adj}(A)$ is $3\times 3$, so determinant scales by $4^{3}=64$.
$|\text{adj}(2A)|=64\,|\text{adj}(A)|=64\times 4=256$.