TITLE: Why is reduced echelon form of a matrix always the identity matrix
QUESTION [7 upvotes]: If $A$ is $3\times3$ matrix with linearly independent columns, then why is the reduced echelon form the identity matrix?

REPLY [2 votes]: Because of two things:

Row operations preserve column dependencies and independencies in matrices.

E.g. if $A=[\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3]$ has the column dependency $\mathbf{a}_1+2\mathbf{a}_2=\mathbf{a}_3$, and $B=[\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3]$ is row-equivalent to $A$, then $\mathbf{b}_1+2\mathbf{b}_2=\mathbf{b}_3$ also.
To prove this, we just check it holds each of the row operations.

The identity matrix is the only matrix in reduced row echelon form with linearly independent columns.

In any other reduced row echelon form matrix, any non-zero column without a leading entry can be written as a linear combination of other columns (a zero column is linearly dependent in itself).