Echelon Form

A matrix is in row echelon form (or just “echelon” form) iff:

• All rows having only zero entries are at the bottom.
• For all row that does not contain entirely zeros, the first non-zero entry is 1.
• For 2 successive non-zero rows, the leading 1 in the higher row is further left than the leading 1 in the lower row.

The process of reducing the augmented matrix to row Echelon form is known as Gaussian elimination.

Column echelon form

A matrix $A$ is in column echelon form if $A^{\text{T}}$ is in row echelon form.