Question:
If $A$ is an invertible matrix of order 3 and $|A|=3$, then $|a d j A|=$___________
Solution:
Given:
$A$ is an invertible matrix of order 3
$|A|=3$
As we know,
$|\operatorname{adj} A|=|A|^{n-1}$, where $n$ is the order of $A$
$\Rightarrow|\operatorname{adj} A|=|A|^{3-1} \quad(\because$ Order of $A$ is 3$)$
$\Rightarrow|\operatorname{adj} A|=|A|^{2}$
$\Rightarrow|\operatorname{adj} A|=(3)^{2} \quad(\because|A|=3)$
$\Rightarrow|\operatorname{adj} A|=9$
Hence, $|\operatorname{adj} A|=\underline{9}$.
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