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So I'm to describe the possible forms of a $2\times 2$ matrix in Echelon form. I thought the only way we could have a $2\times 2$ matrix in Echelon form is like this: $$\left[\begin{matrix}1&c\\ 0&0\end{matrix}\right]$$ Where $c$ is just some constant. But I'm told that we can have it in the form $$\left[\begin{matrix}1&c\\0&1\end{matrix}\right]$$ and $$\left[\begin{matrix}0&1\\0&0\end{matrix}\right]$$But how come we can have a leading entry in the last column when this just leaves the matrix inconsistent? Is it because consistency doesn't matter when it comes to just classifying a matrix as "in Echelon form"?

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There is "row echelon form" and "reduced row echelon form"; the stricter of the two is the last.

The linked entry contains the criteria for each, first the row echelon form, then additional criteria for reduced row echelon form. So using the term "echelon form", or even the casual use of the terms "row echelon form", can be ambiguous, depending on context, and depending on textbooks, to some degree, unless an explicit distinction is made.

Also, as an aside, consistency isn't a property of a matrix, per se, but of the system of equations that an augmented coefficient matrix may represent. For your matrices, the first reveals that there exist two linearly independent row (and column) vectors. Your second matrix reveals that the row vectors (and column vectors) are linearly dependent. We can also determine the rank of a matrix from a matrix in echelon form: the rank of a matrix is equal to the number of non-zero rows in the matrix when reduced to row echelon form.

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+1 For a $2\times 2$ matrix it is done too fast, but for higher it takes time. –  B. S. Feb 5 '13 at 3:49
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Kyle - does the link help clarify matters? Do you know what I mean when I say that a matrix itself isn't consistent/non-consistent, but that consistency is a property of the linear system it may represent? (which is usually represented by an augmented coefficient matrix, as in your more recent post?) –  amWhy Feb 5 '13 at 14:41
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