@PeterTamaroff That is where I was at. I denoted $R_1,\cdots,R_i,\cdots, R_j, \cdots, R_n$ the rows of A
Then I considered a linear combination transformation $R_j$ into $R_j+\lambda R_i$
such that $R_1,\cdots,R_i,\cdots, R_j+\lambda R_i, \cdots, R_n$ are the rows of A'