# 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.

**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.

**Reduced.**

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