linear algebra - Would a matrix have more than one reduced echelon matrix if each reduced echelon matrix was achieved using different elementary row operations?

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.

Answer

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.