If A is an invertible matrix of order 3 and

Given:

$A$ is an invertible matrix of order 3

$|A|=4$

As we know,

$|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(n-1)^{2}}$, where $n$ is the order of $A$

$\Rightarrow|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(3-1)^{2}} \quad(\because$ Order of $A$ is 3$)$

$\Rightarrow|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(2)^{2}}$

$\Rightarrow|\operatorname{adj}(\operatorname{adj} A)|=|A|^{4}$

$\Rightarrow|\operatorname{adj}(\operatorname{adj} A)|=4^{4} \quad(\because|A|=4)$

$\Rightarrow|\operatorname{adj}(\operatorname{adj} A)|=256$

Hence, $|\operatorname{adj}(\operatorname{adj} A)|=\underline{256}$.