In this paper we define the incidence matrix of a link diagram via its signed
planar graph and its dual graph. With a recent result of Ayaka Shimizu, we show
that a link diagram has one component if and only if the $\mathbb{Z}_2$-rank of
its incidence matrix exactly equals to the crossing number of the diagram. By
studying the effect of region crossing changes on 2-component link diagram we
show that region crossing change on 2-component link diagram is an unknotting
operation if and only if the linking number of the diagram is even. In general,
we prove that a link diagram represents an $n$-component link if and only if
the $\mathbb{Z}_2$-rank of its associated incidence matrix equals to $c-n+1$,
here $c$ denotes the crossing number of the diagram. Finally some application
of our result with the viewpoint of graph theory are discussed.