Question

To determine whether the given relation is asymmetric from the given matrix \(M\) (of the relation).

Short answer

The given relation is asymmetric if and only if there exist a pair of integers such that the matrix entries\({M_{ij}} \ne {M_{ji}}\).

Step-by-step solution

Given data

The matrix \(M\) of the relation is given.

Concept of Symmetric relation

A relation is symmetric if and only the matrix\(M\)of the relation is symmetric, in other words, for every pair of distinct integers\(i\)and\(j\).

\({M_{ij}} = {M_{ji}}\) …….(1)

Calculation to check whether the given relation is asymmetric or not

From (1) it follows that the relation represented by the matrix is symmetric if and only if all the corresponds off-diagonal entries are equal. Consequently, the relation is asymmetric if there is at least one pair of distinct integers \(i\) and \(j\) such that \({M_{ij}} \ne {M_{ji}}\).

In other words, there is asymmetry about the diagonal of the matrix.