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

Question

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 1

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

Answer 2

Let's think about a square, Row-reduced Echelon matrix. If all the columns have leading 1s, which is equal to all rows having a leading one. Due to shape of the row-reduced echelon matrix it becomes Identity matrix, and you know I*I = I. So inverse of identity do exist.

Now assume that it lacks a leading 1 at least for one column/row . So at least the last row becomes full of 0's. Assume for a contradiction that this row-reduced echelon form with at least one row with full of zeros, have an inverse T. So RT = I But since at least the last row is full of zeros. at least last row of RT is full of zeros as well. So R*T cannot be the identity matrix. This completes the proof, it has to be the identity matrix. there is no other way