Suppose $n>1$ and $A$ is a non-singular matrix of order $n$ such that $\left|adjA\right|=\left|adj\left(adjA\right)\right|$. Then the matrix whose rank is $n$, is

We know that $\left|adjA\right|={\left|A\right|}^{\left(n-1\right)} \& \left|adj\left(adjA\right)\right|={\left|A\right|}^{{\left(n-1\right)}^2}$, where $n$ is the order of the matrix $A$.

Given $\left|adjA\right|=\left|adj\left(adjA\right)\right|$

$\Rightarrow {\left|A\right|}^{\left(n-1\right)}={\left|A\right|}^{{\left(n-1\right)}^2}$

$\Rightarrow \left(n-1\right)={\left(n-1\right)}^2$

$\Rightarrow \left(n-1\right)\left(n-1-1\right)=0$

$\Rightarrow \left(n-1\right)\left(n-2\right)=0$

$\Rightarrow n=1$ (not possible as $n>1$) or $n=2$

So, $n=2$.

Rank of the matrix refers to the highest number of linearly independent columns in a matrix or the highest number of linearly independent rows in the matrix.

So, $\left[\begin{array}{lll}1& 2& 3\\ 4& 5& 6\\ 6& 7& 8\end{array}\right]$ is the required matrix.