When we are doing the Gaussian Elimination, Can we just interchange the rows?

Question

Given: v1 = [2, −1, 2, 0], v2 = [1, 0, 1, 1], v3 = [3, −2, 3, −1], and v4 = [3, 1, 2, 4].

[2, −1, 2,  0]
[1, 0,  1,  1]        
[3, −2, 3, −1]
[3, 1,  2,  4]

Can we just interchange R1 and R2 since the leading element, 1, of [1, 0, 1, 1] is just in the right position? Became something like this:

[1, 0,  1,  1]   
[2, −1, 2,  0]     
[3, −2, 3, −1]
[3, 1,  2,  4]

Answer 1

Yes, we can. If you think in terms of systems of equations, it corresponds to exchanging two of the equations. It changes nothing.

Answer 2

The basic idea in Gaussian elimination is to go through a sequence of trial positions trying to get a 1 in that position and 0’s below if possible. Thus you can interchange the row in the trial position with a row below it under certain circumstances (i) if the element in the trial position and an element below it in the same column are both non-zero, you can interchange those rows if you wish (ii) if the element in the trial position is 0 and there is some non-zero element below it in the same column, you should interchange those rows. After this preliminary step, no more row interchanges are needed for that trial position. If you now have a non-zero element in the trial position, you use pivot operations bring the other elements in that column and below all to 0.