Row Equivalence is Equivalence Relation

Theorem
Row equivalence is an equivalence relation.

Proof
In the following, $\mathbf A$, $\mathbf B$ and $\mathbf C$ denote arbitrary matrices in a given matrix space $\map \MM {m, n}$ for $m, n \in \Z{>0}$.

We check in turn each of the conditions for equivalence:

Reflexive
Let $r_i$ denote an arbitrary row of $\mathbf A$.

Let $e$ denote the elementary row operation $r_i \to 1 r_i$ applied to $\mathbf A$.

Then trivially:
 * $\map e {\mathbf A} = \mathbf A$

and so $\mathbf A$ is trivially row equivalent to itself.

So row equivalence has been shown to be reflexive.

Symmetric
Let $\mathbf A$ be row equivalent to $\mathbf B$.

Let $\Gamma$ be the row operation that transforms $\mathbf A$ into $\mathbf B$.

From Row Operation has Inverse there exists a row operation $\Gamma'$ which transforms $\mathbf B$ into $\mathbf A$.

Thus $\mathbf B$ is row equivalent to $\mathbf A$.

So row equivalence has been shown to be symmetric.

Transitive
Let $\mathbf A$ be row equivalent to $\mathbf B$, and let $\mathbf B$ be row equivalent to $\mathbf C$.

Let $\Gamma_1$ be the row operation that transforms $\mathbf A$ into $\mathbf B$.

Let $\Gamma_2$ be the row operation that transforms $\mathbf B$ into $\mathbf C$.

From Sequence of Row Operations is Row Operation, $\mathbf C$ is row equivalent to $\mathbf A$.

So row equivalence has been shown to be transitive.

Row equivalence has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.

Also see

 * Column Equivalence is Equivalence Relation