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# 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.

MathJax TeX Test Page $$\begin{bmatrix}1 & 0 & 2 & 3 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix}$$

## 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.

MathJax TeX Test Page $\begin{bmatrix}1 & 0 & 2 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{bmatrix}$ Reduced.

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