If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$?

If so, then prove it. If not, then provide a counterexample.

Definition (Row Equivalent).

Two matrices are said to be row equivalent if one can be obtained from the other by a sequence of elementary row operations.

Proof.

Yes, in this case $A$ and $C$ are row-equivalent.

By assumption, the matrices $A$ and $B$ are row-equivalent, which means that there is a sequence of elementary row operations that turns $A$ into $B$.

Call this sequence $r_1 , r_2 , \cdots , r_n$, where each $r_i$ is an elementary row operation.
(Start with applying $r_1$ to $A$.)

By another assumption, $B$ is row-equivalent to $C$, which means that there is a sequence of elementary row operations which transforms $B$ into $C$; call this sequence $s_1 , s_2 , \cdots , s_m$.

Putting these sequences together, the operations $r_1 , r_2 , \cdots , r_n$ , $s_1 , s_2 , \cdots , s_m$ will transform the matrix $A$ into $C$.

This proves that $A$ and $C$ are row-equivalent.

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