Echelon Form

# Echelon Form | Engineering Mathematics - Civil Engineering (CE) PDF Download

### How to Change a Matrix Into its 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.

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.

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.

• Pivot the matrix
• Find the pivot, the first non-zero entry in the first column of the matrix.
• Interchange rows, moving the pivot row to the first row.
• Multiply each element in the pivot row by the inverse of the pivot, so the pivot equals 1.
• 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.
• To get the matrix in row echelon form, repeat the pivot
• Repeat the procedure from Step 1 above, ignoring previous pivot rows.
• Continue until there are no more pivots to be processed.
• To get the matrix in reduced row echelon form, process non-zero entries above each pivot.
• Identify the last row having a pivot equal to 1, and let this be the pivot row.
• Add multiples of the pivot row to each of the upper rows, until every element above the pivot equals 0.
• 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.
To transform matrix A into its echelon forms, we implemented the following series of elementary row operations.

• 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.
• Working with matrix A1, we multiplied each element of Row 1 by -2 and added the result to Row 3. This produced A2.
• Working with matrix A2, we multiplied each element of Row 2 by -3 and added the result to Row 3. This produced Aref. Notice that Aref 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.
• And finally, working with matrix Aref, we multiplied the second row by -2 and added it to the first row. This produced Arref. Notice that Arref 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.

The document Echelon Form | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
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## FAQs on Echelon Form - Engineering Mathematics - Civil Engineering (CE)

 1. What is echelon form in linear algebra?
Ans. Echelon form in linear algebra refers to a specific form that a matrix can be transformed into through a series of row operations. In echelon form, all rows consisting entirely of zeros are at the bottom, and the leading entry (the first non-zero entry) of each row is to the right of the leading entry of the row above it. This form is useful for solving systems of linear equations and performing other operations in linear algebra.
 2. How is echelon form different from reduced echelon form?
Ans. Echelon form and reduced echelon form are two related but distinct forms of a matrix. While echelon form only requires the leading entry of each row to be to the right of the leading entry of the row above it, reduced echelon form imposes additional restrictions. In reduced echelon form, the leading entry of each row is equal to 1, and all other entries in the column containing a leading entry are zero. This additional condition makes the reduced echelon form unique for each matrix.
 3. What are the benefits of using echelon form in linear algebra?
Ans. Echelon form provides several benefits in linear algebra. It simplifies the process of solving systems of linear equations, as it enables us to identify quickly the variables that are leading and those that are free. Additionally, echelon form makes it easier to perform operations such as matrix addition, subtraction, and multiplication. It also helps in finding the rank of a matrix, which is an important determinant in various applications of linear algebra.
 4. How can a matrix be transformed into echelon form?
Ans. A matrix can be transformed into echelon form through a series of row operations. The three primary row operations are: (1) swapping two rows, (2) multiplying a row by a non-zero scalar, and (3) adding a multiple of one row to another row. By applying these operations strategically, we can eliminate all entries below each leading entry, resulting in the desired echelon form.
 5. Can every matrix be transformed into echelon form?
Ans. Yes, every matrix can be transformed into echelon form through a series of row operations. This process is known as row reduction or Gaussian elimination. However, it is important to note that the resulting echelon form may not be unique for a given matrix. Different sequences of row operations can lead to different echelon forms, but they will all share the same essential properties of echelon form as described in the first question.

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