TITLE: Proof that the method of Gauss/Jordan yields the inverse of a matrix
QUESTION [5 upvotes]: I have trouble in solving the following exercise: let A be an invertible matrix. Consider the matrix A|I where I is the identity matrix.
Prove that the matrix obtained by transforming A|I in it's row-echelon form is equal to I|B where B is the inverse matrix of A. Any help would be appreciated.

REPLY [4 votes]: In the matrix $A|I$ imagine we place in the middle the matrix inverse $B$ and we have
$$AB|I$$
which means obviously that $AB=I$ and an elementary row  operation on A is multiplying A from the left  by an elementary matrix $E_j $ until we find the matrix $I$ so we find
$$\underbrace{E_rE_{r-1}\cdots E_1A}_{=I}B|E_rE_{r-1}\cdots E_1I$$
hence we find that
$$B=E_r E_{r-1}\cdots E_1$$
and we can do the same method for the elementary column operations.