Reduced Row-Echelon Form is a form of matrix, where each nonzero entry in a row is 1 and is the only non-zero entry in that column. This form of matrix is mainly used in linear algebra. The word ” echelon ” is actually taken from the French word ” échelon ” which means ‘ level ‘ or ‘ steps of a ladder ‘. A matrix is said to be in reduced row echelon form when it is in row echelon form and has a non zero entry. In this article we will discuss in details about reduced row echelon form, how to transfer matrix to reduced row echelon form and other topics related to it.

What is Echelon Form?

Echelon form of a matrix is used to solve linear equations. It is effective in changing a complex matrix into a simple matrix. A matrix is in echelon form if it satisfies some conditions which we are discussed below:

• The first non zero element in a row is 1. This entry is known as pivot.
• Each pivot in a column is in the right side of the pivot column in a previous row.
• A zero row should be below the non zero rows.

Example:

[Tex]\begin{bmatrix} 1 2 3 & & \\ 0 1 2 & & \\ 0 0 1 & & \\ \end{bmatrix}[/Tex]

[Tex]\begin{bmatrix} 1 0 2 & & \\ 0 0 1 & & \\ 0 0 0 & & \\ \end{bmatrix}[/Tex]

What is Row Echelon Form?

In linear algebra, a matrix is in row echelon form if it is obtained as the result of Gaussian Elimination on the rows of that matrix. This form of matrix must satisfy some conditions which are discussed below:

• The leading non zero element or pivot of every row is at the right of the leading element of the first row.
• Any row consisting entirely of zeros comes at the bottom of the matrix.

Example:

[Tex]\begin{bmatrix} 1 2 0 0 & & & \\ 0 1 0 3 & & & \\ 0 0 1 0 & & & \\ 0 0 0 0 & & & \\ \end{bmatrix}[/Tex]

Reduced Row-Echelon Form with Examples

To understand what is reduced row echelon form, we should first go through the row echelon form. In particular, remember that, a matrix is always in row echelon form if and only if,

• In a matrix, all its non zero rows have at least one non zero entry
• The non zero entry must have zeros below it and to its left.
• zero rows are below the non zero rows.

Example:

[Tex]\begin{bmatrix} 1 6 9 & & \\ 0 0 8 & & \\ 0 0 0 & & \\ \end{bmatrix}[/Tex]

Here, matrix C is in row-echelon form. The first and second rows are non zero rows. The third row is a zero row.

In reduced row echelon form, a matrix must satisfy the following properties:

• The leading non zero entry is 1
• Leading 1 is the only nonzero entry in its column
• The leading 1 in the rows are always right of the leading 1 of the first row
• If there is any rows containing all zeros, they should be at the bottom of the matrix.

Example 1:

[Tex]\begin{bmatrix} 1 0 0 & & \\ 0 1 0 & & \\ 0 0 0 & & \\ \end{bmatrix}[/Tex]

The above matrix is in reduced row-echelon form. The third row is the zero row which is at the bottom of the non zero rows. The first and second row are non zero rows. A₁₁ & A₂₃ are non zero entries and equal to 1.

Example 2:

[Tex]\begin{bmatrix} 1 0 0 & & \\ 0 1 0 & & \\ 0 0 1 & & \\ \end{bmatrix}[/Tex]

The above matrix is in reduced row-echelon form. All rows are non zero rows.

What is Gaussian Elimination?

Gaussian elimination is a process of converting a matrix into reduced row echelon form through various steps. This process is indeed applied on rows or columns of a matrix. When it is applied on the rows, it is called row echelon form and when it is applied on the column, it is called column echelon form. It is used to find a solution to the problems of linear equations. The steps or operations of Gaussian Elimination are as following:

• Interchange any two rows
• Add or Subtract two rows together
• Multiply one row by a non zero constant.

Example of Gaussian Elimination

Solve the following system of equations

x + y + z = 2

x + 2y + 3z = 7

2x + 3y + 4z = 13

Solution:

The given linear equations are

x + y + z = 2

x + 2y + 3z = 7

2x + 3y + 4z = 13

The augmented matrix of these linear equations is

1 1 1 : 2

1 2 3 : 7

2 3 4 : 13

Now subtracting R₁ from R₂, i.e., (R₂=R₂-R₁)

1 1 1 : 2

0 1 2 : 5

2 3 4 : 13

Now subtracting 2R₁ from R₃, i.e., (R₃=R₃-2R₁), we get,

1 1 1 : 2

0 1 2 : 5

0 1 2 : 9

Now we will subtract R₂ from R₁, after that we will get new elements of R₁

1 0 -1 : -3

0 1 2 : 5

0 1 2 : 9

Lastly we will subtract R₂ from R₃ to get new elements of R₃

1 0 -1 : -3

0 1 2 : 5

0 0 0 : 4

Now putting the values of x, y and z in the equations, we get,

x – y = -3

y + 2z = 5

0 = 4 , which is impossible.

∴The above system of equations have no solutions.

Steps to Transformation of a Matrix to Reduced Row-Echelon Form

The step by step process to convert a matrix to reduced row echelon form are discussed below:

Let us consider a matrix

[Tex]\begin{bmatrix} 1 2 -1 -4 & & & \\ 2 3 -1 -11 & & & \\ -2 0 -3 22 & & & \\ \end{bmatrix}[/Tex]

Now, we will discuss the steps one by one,

1. Make the leading entry of the first row = 1

If there is a 1 in the leading entry of the first row then, no change is required in this step. We have 1 in the leading entry in the given matrix A.

2. Eliminate the other entries in the first column

• First we will make changes in the second row of the matrix: R2 = R2 – 2R1
• Then, for the third row: R3 = R3 + 2R1

1 2 -1 -4

A = 0 -1 1 -3

0 4 -5 14

3. Make the leading entry of the second row = 1

After the first row and column, we have to make the leading entry of the second row = 1 : R2 = – R2

1 2 -1 -4

A = 0 1 -1 3

0 4 -5 14

4. Eliminate the other entries in the second column

• First we will make changes in the first row: R1 = R1 – 2R2
• For the third row: R3 = R3 – 4R2

1 0 1 -10

A = 0 1 -1 3

0 0 -1 2

5. Make the leading entry of the third row = 1

Now we have to make the leading entry of the third row = 1.

Therefore, R3 = -R3

1 0 1 -10

A = 0 1 -1 3

0 0 1 -2

6. Eliminate the other entries in the third column

• First we will make changes in the first row: R1 = R1 – R3
• then, for the second row: R2 = R2 + R3

1 0 0 -8

A = 0 1 0 1

0 0 1 -2

Finally, we get the reduced row echelon form of the matrix A

1 0 0 -8

A = 0 1 0 1

0 0 1 -2

Solved Problems on Reduced Row Echelon Form

Find the reduced row echelon form of the given matrix.

3 −2 −3 3

2 3 3 2

Solution:

The given matrix is

3 −2 −3 3

2 3 3 2

Now, R1 – R2 → R1

1 −5 −6 1

2 3 3 2

Now, R2 – 2R1 → R2

1 −5 −6 1

0 13 15 0

Now, 1/13 R2 → R2

1 −5 −6 1

0 1 15/13 0

At the end, R1 + 5R2→ R1

1 0 -3/13 1

0 1 15/13 0

∴ The row reduced echelon form of the given matrix is

1 0 -3/13 1

0 1 15/13 0

Practice Questions on Reduced Row Echelon Form

Q1. Solve the following system of equations using gaussian Elimination.

2x – 2y = -6

x – y + z = 1

3y – 2z = -7

Q2. Solve this system of equations using gaussian Elimination.

x + y = 5

3x – 2y = 6

Q3. Solve this system of equations using gaussian Elimination.

-x + 4y – 2z = -15

-4x + 6y + z = – 5

-6x – 6y – 2z = – 10

Q4. Is the following matrix in reduced row echelon form? explain.

5 0 0 7

0 1 0 5

0 0 1 4

Q5. Solve the following system of equations.

x + y = 2

3x + 4y = 5

4x + 5y = 9

Reduced Row-Echelon Form- FAQs

What is zero row echelon form?

0 0

0 0

The above matrix is a zero matrix or a null matrix. The zero matrix is row equivalent only to itself, hence it can be said that a zero matrix is its own RREF.

What is the difference between echelon and reduced echelon?

The difference between the echelon and reduced echelon form is that, the matrix in echelon form can contain any values as its non-zero entries, while the one in reduced echelon form can only contain ones.

What is reduced row echelon form or RREF used for?

Reduced row echelon form of a matrix is used to solve system of linear equations.

Is reduced row echelon form identity matrix?

1 0 0

0 1 0

0 0 1

The above 3✖3 matrix is in the reduced row echelon form and is also an identity matrix. Hence we can say that RREF can form an identity matrix.

How many solutions does RREF have?

When the number of variables are more than the number of nonzero rows in the reduced row echelon form of a matrix, the matrix equation will have infinitely many solutions.