<s>[INST] To find the inverse of a special kind of matrix. In a matrix analysis problem, I encountered the following special kind of matrix
$$  
    \begin{bmatrix}
    0 & 1 & a & a & a & a \\
    1 & 0 & a& a& a& a \\
    a& a &0 & 1& a& a \\
    a& a &1 & 0 & a& a \\
    a & a & a & a &0 & 1\\
    a & a & a & a &1 & 0
    \end{bmatrix}
$$
where $a$ is a positive integer.But this matrix is not a circulant matrix. It is not in my knowledge if this is any known form.
We can also search for inverses for the general form of the matrix for even order.
 [/INST] You can write it as a sum:
$$ \begin{bmatrix}
0 & 1 & a & a & a & a \\ 
1 & 0 & a & a & a & a \\ 
a & a & 0 & 1 & a & a \\ 
a & a & 1 & 0 & a & a \\ 
a & a & a & a & 0 & 1 \\ 
a & a & a & a & 1 & 0 
\end{bmatrix} = 
\begin{bmatrix}
-a & 1-a & 0 & 0 & 0 & 0 \\ 
1-a & -a & 0 & 0 & 0 & 0 \\ 
0 & 0 & -a & 1-a & 0 & 0 \\ 
0 & 0 & 1-a & -a & 0 & 0 \\ 
0 & 0 & 0 & 0 & -a & 1-a \\ 
0 & 0 & 0 & 0 & 1-a & -a \end{bmatrix}
+
\begin{bmatrix} a \\ a\\ a \\ a \\ a \\ a \end{bmatrix}.\begin{bmatrix}1 \\ 1 \\ 1 \\ 1\\ 1 \\ 1 \end{bmatrix}^T $$
 and then use the Sherman–Morrison formula to invert this sum of two matrices. The dimension of the matrix can be easily adapted.
</s>