Why Row operation does not change the column rank?

If a collection of columns are linearly independent (respectively linearly dependent) then they remain so under elementary row operations. Therefore elementary row operations do not change the largest set of linearly independent columns.

To see this, notice that a linear dependence relation between some columns of a matrix $A$ is given by a nonzero column vector $v$ with $Av=0$. If $E$ is an elementary matrix and $B=EA$ then $Bv=EAv=0$; conversely if $Bv=0$ then $Av=E^{-1}Bv=0$. Therefore if a set of columns of $A$ is linearly dependent then the corresponding set of columns of $B$ is also linearly dependent and vice versa. This means $A$ and $B$ have the same column rank.