What do you mean by "given row"? In row echelon form, only the first row can have the form "1 0 0 0 0" since all columns before the "pivot" (the main diagonal) must be 0.b for this particular row is a non zero value.
A matrix has rank less than its size if and only if, in row echelon form, there are one or more rows that consist entirely of 0's. The matrix equation then will have a solution (and, in fact, an infinite number of solutions) if and only if the corresponding rows in the reduce form of b are also 0: That is the case when "rank A = rank [A|b] < n" since "rank A" is the number of non 0 rows in row echelon form and "rank[A|b]" is the number of non 0 rows in the row echelon form of the augmented matrix.
What do you mean by "maintain" the rank"?Wouldnt this still maintain the rank however make it so there is no solution?
Also is there a way to solve this problem mathematically? My proof has pretty much been all words which I dont find to be a very concrete answer.