Summary - AP statistics Matrix algebra tutorial-Echelon Form of a Matrix

Echelon Form of a Matrix
This lesson introduces the concept of an echelon matrix. Echelon matrices come in two forms: the row echelon form (ref) and
the reduced row echelon form (rref).


Row Echelon Form
A matrix is in row echelon form (ref) when it satisfies the following conditions.


The first non-zero element in each row, called the leading entry, is 1.

Each leading entry is in a column to the right of the leading entry in the previous row.

Rows with all zero elements, if any, are below rows having a non-zero element.


Each of the matrices shown below are examples of matrices in row echelon form.

1 2 3 4
1 2 3 4 1 2
0 0 1 3
0 0 1 3 0 1
0 0 0 1
0 0 0 1 0 0
0 0 0 0
Aref Bref Cref

Note: Some references present a slightly different description of the row echelon form. They do not require that the first non-zero ent
in each row is equal to 1.


Reduced Row Echelon Form

A matrix is in reduced row echelon form (rref) when it satisfies the following conditions.


The matrix satisfies conditions for a row echelon form.

The leading entry in each row is the only non-zero entry in its column.


Each of the matrices shown below are examples of matrices in reduced row echelon form.

1 2 0 0
1 2 0 0 1 0
0 0 1 0
0 0 1 0 0 1
0 0 0 1
0 0 0 1 0 0
0 0 0 0
Arref Brref Crref


Test Your Understanding
Problem 1

Which of the following matrices is in row echelon form?


0 1 1 2 1 2 1 0
1 0 0 1 0 1 0 0