Why is a negative sign not needed when swapping rows in matrix row echelon form?

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Swapping rows in the context of calculating determinants requires a negative sign to maintain the value of the determinant. However, when reducing a matrix to row echelon form, the operation results in a new matrix, and there is no need to adjust for the determinant's value. The two processes serve different purposes; row reduction aims to simplify the matrix rather than preserve any determinant properties. Therefore, the negative sign is only relevant in the context of determinants, not in row echelon form transformations. Understanding this distinction clarifies why a negative sign is not needed when swapping rows in row echelon form.
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Hi..I have a very basic query...while solving a determinant, when we exchange/swap 2 rows we need to add a negative sign to the determinant. However, when we are trying to reduce a matrix to a row echelon form, when we swap 2 rows..do we need to add a negatice sign here as well? Well..from what I've read...theres no need to..But I'm not sure why..can anyone throw some more light on this??..thanks
 
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You are confusing two completely different things.

When you "row reduce" to find a determinant, you want to keep the result the same- the determinant.

When you row reduce to row-echelon form, you are getting a completely new matrix- you don't have to do anything to "keep them the same" because they aren't supposed to be "the same".
 
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