Echelon matrices

Definition: A matrix is in row echelon form if:

1. All nonzero rows are above any zero rows
2. The leading entry (pivot) of each nonzero row is to the right of the leading entry of the row above it
3. All entries below each pivot are zero

A matrix is in reduced row echelon form (RREF) if additionally:
4. Each pivot is 1
5. Each pivot is the only nonzero entry in its column

Example (row echelon form):
$$\begin{pmatrix} 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

Example (RREF):
$$\begin{pmatrix} 1 & 0 & 2 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

Echelon form is obtained through Gaussian elimination. The number of nonzero rows equals the rank of the matrix. Every matrix has a unique RREF.