If Two Matrices Have the Same Rank, Are They Row-Equivalent?

Problem 644

If $A, B$ have the same rank, can we conclude that they are row-equivalent?

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

Solution.

Having the same rank does not mean they are row-equivalent.

For a simple counterexample, consider $A = \begin{bmatrix} 1 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \end{bmatrix}$.

Both of these matrices have rank 1, but are not row-equivalent because they are already in reduced row echelon form.

Another solution.

The problem doesn’t specify the sizes of matrices $A$, $B$.

Note that if the sizes of $A$ and $B$ are distinct, then they can never be row-equivalent.
Keeping this in mind, let us consider the following two matrices.

$A=\begin{bmatrix} 1 \\ 0 \end{bmatrix} \text{ and } B=\begin{bmatrix} 1 & 0 \end{bmatrix}.$ Then both matrices are in reduced row echelon form and have rank $1$.
As noted above, they are not row-equivalent because the sizes are distinct.