Let there be $A_n=\left(\begin{matrix}4&2&\cdots&2\\2&4&\ddots&\vdots\\\vdots&\ddots&\ddots&2 \\2&\cdots&2&4\end{matrix}\right)\in M_n\left(\mathbb{Z}_7\right)$
I came to the conclusion that on the main diagonal there is $(4,-8,16,-32,...)$ or in $mod(7)$ $(4,-1,2,-4,1,...,)$. I am trying to find the general formula for $det(A_n)$
$A_n=2(1)+2I$, where $(1)$ is the matrix of ones and $I$ is the identity.
Let us study $\det(A(t))=\det((1)-tI)$. The eigenvalues of $(1)$ are $t=0$ with multiplicity $n-1$ and $n$ with multiplicity $1$. Therefore $\det(A(t))=(-1)^n(t^n-nt^{n-1})$.
Therefore $\det(A_n)=2^n(-1)^n((-1)^n-n(-1)^{n-1}))$.
Now simplify at will.