How do you determine the difference between a row echelon matrix and a row reduced echelon matrix just from looking at it. I looked at the conditions and the examples at paul's online math notes and they looked pretty similar with the exception of a couple of numbers not being reduced in the matrices?
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A row echelon matrix has the requirements of having a sort of diagonal from the top left to the bottom right. Everything below the pivots of the diagonal is a 0. A row reduced echelon matrix has everything a row echelon matrix, but also that the pivots are all 1 and all the numbers above each pivot is a 0.
So you mean that each number in the diagonal of a reduced row echelon matrix has a zero above and below it?
Well, in the reduced one, the pivot is a 1, and the entries above and below the pivot are all zeroes. Keep in mind that if the "diagonal" skips a column, it is possible that the numbers in the skipped column have nonzero numbers in it.