How to Change a Matrix Into its Echelon Forms

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.

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

• The matrix is in row echelon form (i.e., it satisfies the three conditions listed above).
• The leading entry in each row is the only non-zero entry in its column.

A matrix in echelon form is called an echelon matrix. Matrix A and matrix B are examples of echelon matrices.

1 2 3 4 0 0 1 3 0 0 0 1 0 0 0 0		1 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0
A		B

Matrix A is in row echelon form, and matrix B is in reduced row echelon form.

How to Transform a Matrix Into Its Echelon Forms

Any matrix can be transformed into its echelon forms, using a series of elementary row operations. Here's how.

1. Pivot the matrix
   1. Find the pivot, the first non-zero entry in the first column of the matrix.
   2. Interchange rows, moving the pivot row to the first row.
   3. Multiply each element in the pivot row by the inverse of the pivot, so the pivot equals 1.
   4. Add multiples of the pivot row to each of the lower rows, so every element in the pivot column of the lower rows equals 0.
      
      
2. To get the matrix in row echelon form, repeat the pivot
   1. Repeat the procedure from Step 1 above, ignoring previous pivot rows.
   2. Continue until there are no more pivots to be processed.
      
      
3. To get the matrix in reduced row echelon form, process non-zero entries above each pivot.
   1. Identify the last row having a pivot equal to 1, and let this be the pivot row.
   2. Add multiples of the pivot row to each of the upper rows, until every element above the pivot equals 0.
   3. Moving up the matrix, repeat this process for each row.

Transforming a Matrix Into Its Echelon Forms: An Example

To illustrate the transformation process, let's transform Matrix A to a row echelon form and to a reduced row echelon form.

0 1 2 1 2 1 2 7 8	⇒	1 2 1 0 1 2 2 7 8	⇒	1 2 1 0 1 2 0 3 6
A		A1		A2

⇒	1 2 1 0 1 2 0 0 0	⇒	1 0 -3 0 1 2 0 0 0
	Aref		Arref

To transform matrix A into its echelon forms, we implemented the following series of elementary row operations.

1. We found the first non-zero entry in the first column of the matrix in row 2; so we interchanged Rows 1 and 2, resulting in matrix A₁.

2. Working with matrix A₁, we multiplied each element of Row 1 by -2 and added the result to Row 3. This produced A₂.

3. Working with matrix A₂, we multiplied each element of Row 2 by -3 and added the result to Row 3. This produced A_ref. Notice that A_ref is in row echelon form, because it meets the following requirements: (a) the first non-zero entry of each row is 1, (b) the first non-zero entry is to the right of the first non-zero entry in the previous row, and (c) rows made up entirely of zeros are at the bottom of the matrix.

4. And finally, working with matrix A_ref, we multiplied the second row by -2 and added it to the first row. This produced A_rref. Notice that A_rref is in reduced row echelon form, because it satisfies the requirements for row echelon form plus each leading non-zero entry is the only non-zero entry in its column.

Note: The row echelon matrix that results from a series of elementary row operations is not necessarily unique. A different set of row operations could result in a different row echelon matrix. However, the reduced row echelon matrix is unique; each matrix has only one reduced row echelon matrix.

Test Your Understanding

Problem 1

Consider the matrix X, shown below.

X  =	0 1 1 2 0 5

Which of the following matrices is the reduced row echelon form of matrix X ?

0 1 1 0 0 0	1 0 0 1 0 0	1 0 0 1 0 1	1 0 0 0 0 1
A	B	C	D

(A) Matrix A
(B) Matrix B
(C) Matrix C
(D) Matrix D
(E) None of the above

Solution

The correct answer is (B). The elementary row operations used to change Matrix X into its reduced row echelon form are shown below.

0 1 1 2 0 5	⇒	1 2 0 1 0 5	⇒	1 2 0 1 0 0	⇒	1 0 0 1 0 0
X		X1		X2		Xrref

To change X to its reduced row echelon form, we take the following steps:

1. Interchange Rows 1 and 2, producing X₁.
2. In X₁, multiply Row 2 by -5 and add it to Row 3, producing X₂.
3. In X₂, multiply Row 2 by -2 and add it to Row 1, producing X_rref.

Note: Matrix A is not in reduced row echelon form, because the leading entry in Row 2 is to the left of the leading entry in Row 3; it should be to the right. Matrix C is not in reduced row echelon form, because column 2 has more than one non-zero entry. And finally, matrix D is not in reduced row echelon form, because Row 2 with all zeros is followed by a row with a non-zero element; all-zero rows must follow non-zero rows.