Row-Echelon Form

What is Row-Echelon Form?

A matrix is in row-echelon form if it has three (basically two) properties:

All nonzero rows are above any rows of all zeros.

If there are any rows with all zeros, they must be at the bottom.

Each leading entry* of a row is in a column to the right of the leading entry of the row above it.

Leading entry means the leftmost nonzero entry in a row. This is also called the pivot. So, the leading entry must be to the right of the leading entry of the row above it. Note, this is considered a matrix in row-echelon form. The leading entry in the second row isn't directly to the right of the leading entry in the first row, but that's okay.

$$\begin{bmatrix}1 & 0 & 2 & 3 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix}$$

All entries below a leading entry must be 0.

This is just to reiterate that the lines are basically in order of their leading entires. 

Reduced Row-Echelon Form

In addition to being in row-echelon form, these matrices have two additional properties.

Each leading entry (in a nonzero row) must be 1.

Refer back to the example of a row-echelon matrix above. There, a leading entry could have been any (nonzero) number. Here, they must be 1. 

The leading entry is the only nonzero entry in the column.

Once again, refer back to the last example. There, you can see there's a 2 above the leading entry in the second row. This is fine for row-echelon form, but it wouldn't work for reduced row-echelon form. Here is an example of a matrix in reduced row-echelon form.

$\begin{bmatrix}1 & 0 & 2 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix}$ Reduced.

Note that although there are 2 elements in the third column, there is no pivot in the third column, so it still works.