Suppose that $A$, $B$ and $C$ are $m\times n$ matrices. Use the definition of row-equivalence (Definition REM) to prove the following three facts.

1. $A$ is row-equivalent to $A$.
2. If $A$ is row-equivalent to $B$, then $B$ is row-equivalent to $A$.
3. If $A$ is row-equivalent to $B$, and $B$ is row-equivalent to $C$, then $A$ is row-equivalent to $C$.

A relationship that satisfies these three properties is known as an equivalence relation, an important idea in the study of various algebras. This is a formal way of saying that a relationship behaves like equality, without requiring the relationship to be as strict as equality itself. We will see it again in Theorem SER.