Determining if logical matrix represents a partial order

[sciencectn]

If we represent a relation using a logical matrix (so that if (a,b) is present in the relation then the element at row a and column b in the matrix is a 1), is there any way to determine that this matrix represents a partial order using matrix operations?

For example, if we have the relation {(1,1),(2,2),(3,3),(1,2)} represented by this matrix:

$$\left( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$$

Is there some way, using matrix operations, to determine if this represents a partial order?

 

[Abdullah Naeem]

Actually, yes. This is a convenient way. For instance, see Mathematics of Fuzziness -- Basic Issues by Xuzhu Wang, Da Raun and Etienne E. Kerre.