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The Reduced Row Echelon Form of a Matrix Is Unique: A Simple Proof

One of the most simple and successful techniques for solving systems of linear equations is to reduce the coefficient matrix of the system to reduced row echelon form. This is accomplished by applying a sequence of elementary row operations (see, e.g., [1, p. 5]) to the coefficient matrix until a matrix B is obtained which satisfies the following description: If a row of B does not consist entirely of zeros then the first nonzero number in the row is a 1 (usually called a leading 1). If there are any rows that consist entirely of zeros, they are grouped together at the bottom of B. In any two successive non-zero rows of B, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row. Each column of B that contains a leading 1 has zeros everywhere else. The matrix B is said to be in reduced row echelon form.