Row-echelon form

A matrix is in row-echelon form if

1. all the zero rows are grouped at the bottom; and

2. each pivot lies farther to the right than the pivots of higher rows.

A pivot is the first nonzero entry of a row.

Gaussian elimination

Gaussian elimination is an algorithm for converting a matrix into row-echelon form through a series of row operations.

1. Find the first column with a nonzero entry from the left

2. If that row is not first, switch it with the first row.

3. Make all the other entries in that column zero by adding a multiple of the first row to the other rows, as needed.

4. Continue these steps from the second row on until the matrix is in row-echelon form.

Reduced row-echelon form

A matrix is in reduced row-echelon form if

1. it is in row-echelon form,

2. each pivot is a $1$, and

3. in each pivot column, each entry other than the pivot is $0$.