Show that the node$-$edge incidence matrix of any $($undirected$)$ cycle of odd

length has determinant equal to $\backslash$pm $2.$ After a permutation of labels, you can take the cycle to

have edges e$1,...,$ e$2$n$+1$ where e$1=\{$v$1,$ v$2\},...,$ e$2$n $=\{$v$2$n$,$ v$2$n$+1\}$ and e$2$n$+1=\{$v$2$n$+1,$ v$1\}$