Row Equivalent Matrices

Last updated Nov 1, 2022

Let $F$ be a Field, $m,n \in \mathbb{N}$. Let $r$ be the Relation so that $ArB$ for $A, B \in F^{m \times n}$ if there exists Elementary Row Operations $e_{1}, \dots , e_{k}$ for some $k \in \mathbb{N}$ so that $$B = e_{k}(e_{k-1} ( \cdots e_{1}(A) \cdots )).$$ Then $r$ is an Equivalence Relation. We call $A, B$ Row Equivalent Matrices if $ArB$. If $A, B$ are Row Equivalent Matrices, then we write $A \sim_{R} B$.

We check the criteria for being an Equivalence Relation:

1. $ArA$ because the identity is an Elementary Row Operation.
2. If $ArB$, then because Elementary Row Operations are Invertible by other Elementary Row Operations, we have that $BrA$.
3. If $ArB$ and $BrC$, then composing the Elementary Row Operations that from $A$ to $B$ with those that go from $B$ to $C$ give us $ArC$. $\blacksquare$