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 $4$).

There is a paper called "On inverting circulant matrices" by S.R. Searle, doi 10.1016/0024-3795(79)90007-7. It is related to your question, but I am not educated enough to condense it to a simple formulae for your precise case. Moreover it has three nonzero elements in each row column.
