# How can all matrices have a _unique_ reduced row echelon form?

Wikipedia states (and I suspect the answer to my question could be that wikipedia should never have been my source) that

every matrix has a unique reduced row echelon form.

I am wondering how this can possibly be a unique matrix when any nonsingular matrix is row equivalent to the identity matrix, which is also their reduced row echelon form.

I am considering two possible interpretations that would result in Wikipedia's statement holding, which are (1) two row equivalent matrices are considered the same matrix$$^1$$ or (2) when Wikipedia says unique they mean only a single, rather than different from all others$$^2$$.

Because I am only a sophomore it seemed unwise for me to guess by myself which, if any, of these explanations would be the correct one, which is why I am once again asking for your help: what is meant when Wikipedia states "every matrix has a unique reduced row echelon form."

$$^1$$ if this were so than the conclusion that all matrices have a unique reduced row echelon form would be trivial since matrixes that have the same reduced row echelon form would just be defined as being the same matrix

$$^2$$ this would imply that they mean that for every matrix there exists one and only one reduced row echelon form, rather than two matrices have the same row echelon form if and only if they are the same matrix

• The answer is (2) because that's what the word unique means. (1) is based on an erroneous understanding of the word unique.
– bof
Nov 17, 2020 at 15:16
• @bof It is important to acknowledge that this usage of unique disagrees with colloquial usage. For example, in the sentence "each user has a unique ID number", it would be understood that every pair of users has distinct ID numbers. It would be very strange to interpret this sentence as meaning only that "each user has exactly one ID number". Nov 17, 2020 at 15:35
• @BenGrossmann What you call "colloquial usage" seems to be jargon of semiliterate computer people. The first time I noticed this absurd usage, back in the 80s, was in a spellcheck program which told me it had checked X words and Y "unique words". This nonstandard usage is especially to be avoided in mathematical contexts, where the word "unique" is often used to mean "unique" (as in the instance which confused the OP), so it's a poor idea to press the word "unique" into service as a replacement for perfectly good English words such as "different" and "distinct".
– bof
Nov 18, 2020 at 1:50
• @bof I find that the "absurd" usage agrees more with the common sense interpretation of the word unique. The word "unique" means "different from all other things". In the "correct" interpretation of "every matrix $A$ has a unique rref form", the idea is that there is one matrix that is "unique" in that it is the rref of $A$. In the alternative interpretation, the idea is that this rref matrix is unique among all matrices that are rref matrices (of some other matrix $A'$). Neither interpretation is more "obvious" from the definition of "unique" alone; both interpretations assume context Nov 18, 2020 at 14:18
• What's the source for that definition of "unique"? I looked it up in the online OED. The 1st definition is "Of which there is only one; single, sole, solitary." The computer-jargon usage which is confusing the OP is the OED's 4th defination with earliest cite from 1995: "Designating a distinct individual who visits a particular website or web page within a specified period of time, who is counted only once for the purposes of visitor statistics regardless of the number of visits they make in that time." In a math context the default meaning is "one and only one".
– bof
Nov 18, 2020 at 15:11

For every matrix $$A$$, there exists exactly one matrix $$B$$ such that $$A$$ is row-equivalent to $$B$$ and $$B$$ is in reduced row-echelon form (rref).
As an example, consider the matrices $$A_1 = \pmatrix{1&2\\0&3}, \quad A_2 = \pmatrix{5&-1\\-1 & 7}, \quad I = \pmatrix{1&0\\0&1}.$$ Both $$A_1$$ and $$A_2$$ are invertible, so $$I$$ is the rref of both of these matrices. $$I$$ is the only matrix that is in rref and row-equivalent to $$A_1$$, so it is the rref of $$A_1$$. $$I$$ is also the only matrix that is in rref and row-equivalent to $$A_2$$, so it is the rref of $$A_2$$. The fact that $$A_1$$ and $$A_2$$ have the same rref does not contradict the fact that they have exactly one rref, i.e. "a unique" rref.