What is wrong with the following proof that every matrix with at least two rows is row

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What is wrong with the following "proof" that every matrix with at least two rows is row equivalent to a matrix with a zero row?
Perform R2 + R1 and R1 + R2. Now rows 1 and 2 are identical. Now perform R2 - R1 to obtain a row of zeros in the second row.
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