When you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon?
[Math] What does echelon mean
linear algebramatricesterminology
Related Solutions
$$\begin{array}{ccc|c} 3&0&-2&-3\\ -2&0&1&-2\\ 0&0&-1&2 \end{array}$$ Multiply the first row by 1/3 to put a pivot at $1,1$:
$$\begin{array}{ccc|c} 1&0&-2/3&-1\\ -2&0&1&-2\\ 0&0&-1&2 \end{array}$$
Add 2 times first row to the second row to clear out the pivot column:
$$\begin{array}{ccc|c} 1&0&-2/3&-1\\ 0&0&-1/3&0\\ 0&0&-1&2 \end{array}$$
Multiply the 3rd row by -1 to put a pivot at 3,3:
$$\begin{array}{ccc|c} 1&0&-2/3&-1\\ 0&0&-1/3&0\\ 0&0&1&-2 \end{array}$$
Clear out third pivot column, first add 2/3 of the 3rd row to the first:
$$\begin{array}{ccc|c} 1&0&0&-7/3\\ 0&0&-1/3&0\\ 0&0&1&-2 \end{array}$$
Second, add 1/3 of the third row to the second to finish clearing the pivot column:
$$\begin{array}{ccc|c} 1&0&0&-7/3\\ 0&0&0&-2/3\\ 0&0&1&-2 \end{array}$$
This is the RRE form of your augmented matrix. Note that your equation never had any solutions from the start, as the RRE indicates on the second row: $0 = -2/3$. Also note that most teachers will probably think that adding extra rows and columns of zeros to a matrix is a mistake (and it is if you don't know why it is ok).
In short, row reduced echelon form(RREF) of a matrix $A$ is such that
i) Every leading entry is 1
ii) Any nonzero rows are above zero rows
iii) any leading entry is strictly to the right of any leading entries above that row
iv) any other entry in a column containing a leading entry is 0 except for the leading entry.
So it does not have to be put in augmented matrix $[A|b]$ to get a RRE form. You are comparing RRE form of matrix $A$ and $[A|b]$.
To see why the statement is true, suppose that you put the matrix $[A|b]$ into RRE form, so you have a matrix E. If E contains a leading entry in its last column, in terms of system of equations, what does it say? And what is the condition for E to not have any leading entry in last column?
Note: If RRE form of $[A|b]$ does contain a leading entry, then it is different from that of $A$. Also, note that RRE form of $[A|b]$ is m by n+1 whereas that of $A$ is m by n.
Solve:
$x+y=1$
$x+y=2$
Then we have
$\ A = \left( {\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} } \right) $
$\ b = \left( {\begin{array}{cc} 1 \\ 2 \end{array} } \right) $
and $Ax=b$
If we turn A into RREF, we get
$\ E = \left( {\begin{array}{cc} 1 & 1 \\ 0 & 0 \end{array} } \right) $
So A has rank 1
and if we put $[A|b]$ into RRE form, we get
$\ E' = \left( {\begin{array}{cc} 1 & 1 & 0 \\ 0 & 0 & 1 \end{array} } \right) $
So augmented matrix has rank 2. Observe what last row says in terms of equations.
Best Answer
As has already been suggested, it has its origins in military vocabulary.
The Wiki article on echelon formation contains many pictures of stuff in echelon formation, and you can immediately see why one might say that the rows of a row-echelon matrix are in "echelon formation." I would have liked to include the wiki images, but for some reason they would not load. I found another one instead: