I'm talking about the row echelon form, not the reduced row echelon form. If it isn't can you give me some examples?
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$\begingroup$This matrix is already in row echelon form: $$ \begin{bmatrix} 1 & 1 \\ 0 & 1 \\ \end{bmatrix} $$ but we can apply the row operation $R_1 \gets R_1 - R_2$ which gives another row echelon form $$ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}. $$
Reduced row echelon forms are unique, however.
$\endgroup$ $\begingroup$No. For example, take $$A=\pmatrix{1&2\cr3&4\cr}\ .$$ Row reduction, first method: $$A\sim\pmatrix{1&2\cr0&-2\cr}\ .$$ Second method: $$A\sim\pmatrix{3&4\cr1&2\cr}\sim\pmatrix{3&4\cr0&\frac23\cr}\ .$$ Different echelon forms.
$\endgroup$ $\begingroup$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.
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