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In the text for linear algebra by David C Lay following is given.

Definition: A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A.

After this an example is given in which a matrix in its initial form is shown and a 1 at first row first column is marked as "pivot".

We still haven't reduced the matrix so how can we know which position is pivot?

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  • $\begingroup$ It suffices to reduce the matrix in row echelon form since every echelon form has same pivot position as the fully reduced one. $\endgroup$ – Li Chun Min Apr 6 '17 at 6:13
  • $\begingroup$ @LiChunMin,the example matrix is not in echelon form $\endgroup$ – Vikram Apr 6 '17 at 6:52
  • $\begingroup$ really? Do not confuse row echelon form (ref) with reduced row echelon form (rref). I am looking at the 4th edition…In example 1, all of them in ref and two of them in rref. In example 2, the matrix is not in ref but he is reducing it to ref. $\endgroup$ – Li Chun Min Apr 6 '17 at 7:05
  • $\begingroup$ @LiChunMin,in example 2 in the first step itself he has marked a pivot position and not all entries below it are zero $\endgroup$ – Vikram Apr 6 '17 at 7:10
  • $\begingroup$ you are so diligent to look at the details. I dunno…perhaps he is trying to suggest that if the first column is non-zero, then it is must be a pivot column … otherwise can you suggest some entries below that make the a11 not to be pivot position? I can't so I think that is the hidden message behind. $\endgroup$ – Li Chun Min Apr 6 '17 at 9:04
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I think it is the position at least one position to the right of the last pivot position, and then the left most position which when reduced is a one with zeros to the left of it.

If the rows/column (I always forget which is row) above has a one with zeros to the left of it, and that row/column has a leading one, it will remain the same when completely reduced.

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You are supposed to reduce the matrix first before you can identify which are the pivot positions. You are not expected to be able to tell where the pivot positions are without reducing the matrix, and for most matrices this would be very hard to do.

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