Let \(A =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 1
\end{bmatrix}\)
and let
\(B =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0
\end{bmatrix}\).
Are \(A\) and \(B\) row equivalent?
The answer is “No”.
No matter what elementary row operations are applied to \(B\),
the entries in the right-most column will remain 0's.
Therefore, one can never transform \(B\) to \(A\) using elementary
row operations.