Why does row echelon form work for finding the Rank of a matrix?

Question

I have been studying Linear Algebra and came across gauss-elimination method. While it does give the correct answers , I am not able to understand how it works. So, we compare each row with rest of the rows to see if any is dependent on the current row. But how does it identify, if One of the rows in the matrix, is a linear combination of 2 or more other rows??

Answer 1

• This builds upon the result that elementary row operations doesn't change the rank of the matrix. Hence if we reduce the original matrix to its row echelon form, the two matrices share the same rank.

• To see that the non-zero rows are linear independent, suppose r1, r2,...,rk are the non-zero rows (sorted in that order), consider c1r1+...+ckrk = 0. Look at the the position of the leading non-zero entry of r1, for that position, the other entries have value 0 there, hence we conclude that c1 = 0. After that, we can conclude the same argument for c2 and so on and conclude that all f them must be zero and hence the rank is equal to the number of non-zero rows.