# A graph G has an even number of perfect matchings if and only if ∃S⊆V(G);(S≠ϕ) such that all vertices in V(G) are adjacent to an even number of vertices in S. At the moment, all I can see is just that finding determinant of A(G) over F2 reveals the parity of perfect matchings.

A graph $G$ has an even number of perfect matchings if and only if $\mathrm{\exists }S\subseteq V\left(G\right);\left(S\ne \varphi \right)$ such that all vertices in $V\left(G\right)$ are adjacent to an even number of vertices in $S$.
At the moment, all I can see is just that finding determinant of $A\left(G\right)$ over ${F}_{2}$ reveals the parity of perfect matchings.
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shosautesseleol
The answer can be found by regarding an element in the kernel of the adjacency matrix over ${F}_{2}$, see the comments.
Note also that although the determinant of the adjacency matrix modulo $2$ gives the parity of the number of matchings, it is not at all true that the determinant of the adjacency matrix can often be used to determine the number of matchings.