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I’m working on a problem asking me to find a general solution of a matrix with infinite solutions. The ordered pairs I’m getting are much different than what the book says.

The book also says that “Each matrix is row equivalent to one and only one reduced echelon matrix” yet it doesn’t make any sense, since I achieved an acceptable reduced row echelon form through different elementary row operations.

Edit - Maybe the theory is suggesting that each matrix in row echelon form has one unique reduced row echelon form, but since I got a different row echelon form, of course I would get a unique reduced row echelon form. But this doesn’t explain why the textbook doesn’t take this into account.

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The reduced row echelon form is in principle unique. When computations are done in floating point, roundoff error can mess with this somewhat. But the most likely cause of your getting a different form is that someone (either you or the author) made a mistake.

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    $\begingroup$ Right after I looked at your answer, I checked my work and realized I missed a negative sign on one of my numbers. Typical. I was going to add pictures, but I don’t have the reputation for that, so chances are that others would have found the mistake too. Thank you. I’ll mark you as the right answer since you were first. $\endgroup$ – AngoMango Aug 23 '19 at 21:30
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The book is right: you can perform different elementary row operations and you'll always get the same reduced row matrix (you can try on a $3 \times 3$ matrix to convince yourself!). See here for a proof.

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