An Elementary Proof that Reduced Row Echelon form of a Matrix is Unique

Many results in a first course in linear algebra rely on the uniqueness of reduced echelon form of a given matrix. Most textbooks either omit the proof of this important result or use ideas and results in a proof that are not typically available when the result is introduced. For example, the proof [1] is relegated to an appendix and relies on the concept of linear dependence relations among columns, while the proof in [3] uses elementary matrices, permutation matrices, and block matrix calculations. Since the first course in linear algebra is often a first introduction to higher level conceptual thinking in mathematics and the careful use of definitions, theorems, and proofs, it is desirable to have an elementary proof of the uniqueness of reduced echelon form that is accessible to students at the time the fact is presented using only the fact that the solution sets of linear systems represented by row equivalent augmented matrices are the same. Consequently, this proof can be offered as a supplement to the main narrative of the class and might spark some interest among potential future majors. The ideas in this proof are not new (see [4] and [2], for example) but the idea of using augmented matrices in the argument is novel. The proof here is also written specifically for undergraduate students in their first linear algebra class.