Is the row echelon form of a system of equations unique? I'm talking about the row echelon form, not the reduced row echelon form. If it isn't can you give me some examples?
This matrix is already in row echelon form:
$$
\begin{bmatrix}
1 &amp; 1 \\
0 &amp; 1 \\
\end{bmatrix}
$$
but we can apply the row operation $R_1 \gets R_1 - R_2$ which gives another row echelon form
$$
\begin{bmatrix}
1 &amp; 0 \\
0 &amp; 1 \\
\end{bmatrix}.
$$

Reduced row echelon forms are unique, however.
Reduced row-echelon form of a matrix is unique, but row-echelon is not.
You may have different forms of the matrix and all are in row-echelon forms.