Speciality of Pivot columns.

Question

Suppose $A$ is a matrix of order $m\times n$ and I have operated a series of operations(elementary row operations) which give a product $E$,and $EA=R$ the echelon row reduced form.Now echelon form will have some pivot columns of identity matrix of the form $[0,0,...,1,0,...,0]$(Box bracket means column matrix).Now when we operate $E^{-1}$ on the matrix $R$,We get $A$,Now suppose we are calculating $E^{-1}$,it is nothing but $E^{-1}I$,Now notice that those pivot columns are same in both $I$ and $R$,So for those columns,the effect of $E^{-1}$ on $I$ will be same as its effect of $R$ i.e. for those columns operating the inverse of $E$,we get back the columns of $A$.So, $A$ and $E^{-1}$ have the same entries in pivot columns.Is there anything special about it,I mean is it really a thing to notice,or it is a trivial thing that is of no use.