I have a quick question regarding the difference between echelon form and reduced row echelon form (rref). According to my googling these seem to be the same, but to me it seems that the difference between the two is that echelon form only requires the first value of the first row to be 1. The first non-zero value of each row after the first one can be any value as long as it's not 0.

As for reduced row echelon form every first non-zero value of a row has to be 1.

Am I correct with this or am I completely mixing things up? If I'm correct, when would the echelon form be fine and when do I have to use RREF?

Thanks :)


Row-echelon form (REF):

(i) Leading nonzero entry of each row is 1.

(ii) The leading 1 of a row is strictly to the right of the leading 1 of the row above it.

(iii) Any all-zero rows are at the bottom of the matrix.

$ $

Reduced row-echelon form (RREF):

(i) REF.

(ii) The column of any row-leading 1 is cleared (all other entries are 0).

$ $

Note: A given matrix (generally) has more than one row-echelon form; however, for any matrix, the reduced row-echelon form is unique. This uniqueness allows one to determine if two matrices are row equivalent (can one be transformed to the other by a sequence of elementary row operation).

  • $\begingroup$ These rules are something that confuse me. See the RREF here: math.tutorvista.com/algebra/gauss-jordan-method.html. In the 2nd matrix, the 2nd row begins with 3 instead of 1. Also, column 3 has 3 and 1 instead of just 1. Also, elements in the diagonal are not 1. How is this RREF? $\endgroup$ – Nav Jun 5 '17 at 17:02

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