Suppose $S$ is an invertible symmetric matrix with the following property:
For the entry in the $i$th row and $j$th column, if $|i-j|$ is an odd number then $S_{ij} = 0$; if $|i-j|$ is an even number then $S_{ij} \neq 0$.
Is it possible to prove that the inverse matrix $S^{-1}$ also has the above property?