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 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.
• 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.