Relations and Zero One Matrices I was studying but realized that I am having trouble grasping the representations of relations using Zero One Matrices.  For example if I have a set A = {1,2,3} and a relation R = {(1,1),  (1,2), (2,3), (3,1)}.  

What is the resulting Zero One Matrix representation? 
How exactly do I come by the result for each position of the matrix? How to tell if it is reflexive, transitive, antisymmetric or symmetric?
The resulting zero-one representation is the $|A|\times |A|$ matrix $M$ with $M_{ij} = 1$ if $(i,j) \in R$, and $M_{ij} = 0$ if $(i,j) \notin R$. In our case, the matrix is
$$ M = \begin{pmatrix} 1 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 1 &amp; 0 &amp; 0 \end{pmatrix}. $$
Take it as an exercise to prove the following properties:


$R$ is reflexive iff the diagonal of $M$ is all 1s.
$R$ is symmetric iff $M$ is symmetric.
$R$ is antisymmetric iff the off-diagonal entries of $M+M^T$ are 0/1 (but not 2).
$R$ is transitive iff the support of $M^2$ is a subset of the support of $M$, where the support is the set of non-zero entries.