linear algebra - If R is a row reduced echelon matrix and is Invertible then it is Identity matrix

While proving that

if A is invertible then, A is row equivalent to I

Steps done are :

• R be row reduced echelon matrix of A


• Then R=P*A, where P is finite product of elementary matrices


• But elementary matrices are invertible, which implies P is invertible


• Given A is invertible, then R(=P*A) is Invertible


• Then R is Identity matrix

I understood first four steps.Is there proof, if R is row reduced echelon matrix and is invertible then R is identity ? (Don't use determinants, rank)

Answer

If R is an echelon and invertible matrix then R must be Identity matrix(Must be full rank, because it is invertible).