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.

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.
