The inverse of the adjacency matrix of an undirected cycle Is there an expression for $A^{-1}$, where $A_{n \times n}$ is the adjacency matrix of an undirected cycle $C_n$, in terms of $A$? 
I want this expression because I want to compute $A^{-1}$ without actually inverting $A$. As one answer suggests, $A$ is non-invertible for certain values of $n$ (namely when $n$ is a multiple of $4$).

For $n=4$, the matrix in question is $$\pmatrix{0&1&0&1\cr1&0&1&0\cr0&1&0&1\cr1&0&1&0\cr}$$ which is patently noninvertible. 
