By definition, an equivalence relation on a set is a relation satisfying three conditions: (named as indicated)
(i) Each element A of the set is equivalent to itself (Reflexivity).
(ii) If A is equivalent to B, then B is equivalent to A (Symmetry).

(iii) If A is equivalent to B and B is equivalent to C, then A is equivalent to C (Transitivity).

Show that row equivalence of matrices satisfies these three conditions. Show that for each of the three elementary row operations these conditions hold.