Question:
Let $A=\left[a_{i j}\right]$ be a square matrix of order 3 with $|A|=2$ and let $C=\left[c_{i j}\right]$, where $c_{i j}=$ cofactor of $a_{i j}$ in $A$. Then, $|C|=$__________
Solution:
Given:
$|A|=2$
Order of $A=3$
As we know,
$|\operatorname{adj}(A)|=|A|^{n-1} \quad$ where $n$ is the order of $A$
Also, $\operatorname{adj}(A)=C^{T}$
$\Rightarrow|\operatorname{adj}(A)|=\left|C^{T}\right|$
$\Rightarrow|\operatorname{adj}(A)|=|C| \quad\left(\because\left|C^{T}\right|=|C|\right)$
$\Rightarrow|A|^{n-1}=|C|$
$\Rightarrow|C|=|A|^{n-1}$
$\Rightarrow|C|=|A|^{3-1}$ $(\because$ order of $A=3)$
$\Rightarrow|C|=|A|^{2}$
$\Rightarrow|C|=2^{2}$ $(\because|A|=2)$
$\Rightarrow|C|=4$
Hence, $|C|=\underline{4}$.