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.

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix
$$\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$$

Find, correct to the nearest degree, the three angles of the triangle with the given vertices
A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)

Whether f is a function from Z to R if 
a) ${\displaystyle f\left(n\right)=\pm n}$. 
b) ${\displaystyle f\left(n\right)=\sqrt{n^2+1}}$. 
c) ${\displaystyle f\left(n\right)=\frac{1}{n^2-4}}$.

How to write the expression $6^{\frac{3}{2}}$ in radical form?

How to evaluate ${\displaystyle \sin \left(\frac{-5\pi }{4}\right)}$?

What is the derivative of ${\cot }^{2}x$ ?

How to verify the identity: $\frac{\cos \left(x\right)-\cos \left(y\right)}{\sin \left(x\right)+\sin \left(y\right)}+\frac{\sin \left(x\right)-\sin \left(y\right)}{\cos \left(x\right)+\cos \left(y\right)}=0$?

Find $\tan \left(22.5^{\circ }\right)$ using the half-angle formula.

How to find the exact values of $\cos 22.5°$ using the half-angle formula?

How to express the complex number in trigonometric form: 5-5i?

The solution set of $\tan <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}=3\cot <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}$ is

How to find the angle between the vector and $x-$axis?

Find the probability of getting 5 Mondays in the month of february in a leap year.

How to find the inflection points for the given function ${\displaystyle f\left(x\right)=x^3-3x^2+6x}$?

How do I find the value of sec(3pi/4)?