If A is a non-singular matrix of order n. then $|\mathrm{adj} (\mathrm{adj} A\left)\right|$ is 

Let A be an nth order nonsingular matrix $\left|A\right|=k(\ne 0)$ then

$\begin{array}{l}\mathrm{adj}A=\left|A\right|\cdot A^{-1}=k{A}^{-1}\\ \therefore |\mathrm{adj}A|=\left|k{A}^{-1}\right|=k^n\left|A^{-1}\right|=k^n\frac{1}{\left|A\right|}=k^{n-1}\end{array}$

Let $\mathrm{B}=\mathrm{adj} \mathrm{A}.$ Then 

$\begin{array}{l}|\mathrm{adj}B|=|B{|}^{n-1}=|\mathrm{adj}A{|}^{n-1}\\ ={\left(k^{n-1}\right)}^{n-1}=k^{n^2-2n+1}\\ \Rightarrow |\mathrm{adj}(\mathrm{adj}A\left)\right|=|A{|}^{n^2-2n+1}\end{array}$