linear algebra - How to prove that to reduce $B$ to echelon form no row interchanges are needed?

Suppose that to reduce a matrix $A$ to row echelon form are necessary $n$ elementary operations $E_1,...,E_n$. Suppose that $E_{n_1},...,E_{n_k}$ are the permutation operations that are needed. How to prove that to reduce $B=E_{n_k} \cdots E_{n_1}A$ to echelon form no row interchanges are needed?

(my linear algebra book says it, but doesn't demonstrate)

Thanks.