Row Equivalence is Equivalence Relation

Definition
Two matrices $$\mathbf{A} = \left[{a}\right]_{m n}, \mathbf{B} = \left[{b}\right]_{m n}$$ are row equivalent if one can be obtained from the other by a finite sequence of elementary row operations.

Theorem
Row equivalence is an equivalence relation.

Proof
Checking in turn each of the critera for equivalence:

Reflexive
Any matrix is trivially row equivalent to itself: for example, multiply any row by $$1$$ by $$r_i \to 1r_i$$.

Symmetric
Each elementary row operation is reversible.


 * The inverse of $$r_i \to ar_i$$ is $$r_i \to \left({a^{-1}}\right)r_i$$.
 * The inverse of $$r_i \to r_i + ar_j$$ is $$r_i \to r_i - ar_j$$.
 * The inverse of $$r_1 \leftrightarrow r_j$$ is $$r_1 \leftrightarrow r_j$$ (i.e. swap them back again).

So let $$\mathbf{A}$$ be row equivalent to $$\mathbf{B}$$.

Let $$\Gamma$$ be the sequence of elementary row operations that turn $$\mathbf{A}$$ into $$\mathbf{B}$$.

Then for each elementary row operation in $$\Gamma$$, the inverses of those operations can be applied in the reverse order on $$\mathbf{B}$$ to get $$\mathbf{A}$$.

Thus if $$\mathbf{A}$$ is row equivalent to $$\mathbf{B}$$, then $$\mathbf{B}$$ is row equivalent to $$\mathbf{A}$$.

Transitive
Let $$\mathbf{A}, \mathbf{B}, \mathbf{C}$$ be $m \times n$ matrices.

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

Let $$\Gamma_1$$ be the sequence of elementary row operations that turn $$\mathbf{A}$$ into $$\mathbf{B}$$.

Let $$\Gamma_2$$ be the sequence of elementary row operations that turn $$\mathbf{B}$$ into $$\mathbf{C}$$.

It follows that, as $$\Gamma_1$$ and $$\Gamma_2$$ are finite sequences of operations, you can apply the operations in $$\Gamma_1$$ to $$\mathbf{A}$$ and follow them immediately by the operations in $$\Gamma_2$$ to get $$\mathbf{C}$$

Thus $$\mathbf{A}$$ is row equivalent to $$\mathbf{C}$$.