In Gaussian Elimination, we take the corresponding row and zero out the furthest left non-zero position in the rows below in order to clear the respective variable and generate a pivot. We do this by starting with the first row and subtracting (or adding) it to the rows below in order to generate a pivot in the first nonzero column available to us.
Now, this method allows us see whether rows are linearly dependent (a zero row appearing, for example implies linear dependence) and get the rank of a matrix, among other things. However, I have some doubt about the guarantee of the process.
In Gaussian Elimination, we start with the first row and we eliminate it from other rows. What if re-arranged the rows and used a different row to do eliminations with first? I know that because an RREF is unique, we'll achieve the same dependence and rank results as doing Gaussian Elimination on the original matrix. My question is though, why exactly does this occur?
In other words, say we are given a matrix A. Presume that originally we use Row1 to clear say the entries in the Col1. Now, Gauss tells me that swapping rows and instead using Row2 to do this work will produce the same results. Why does this occur?