Echelon Row Reduction

An echelon matrix is a matrix that has been transformed to a specific structured form using row operations. The most common types are row echelon form (REF) and reduced row echelon form (RREF).  The Echelon Row Reduction process is widely used in solving systems of linear equations, performing Gaussian elimination, and analyzing linear algebra properties. Here is an example of echelon row reduction.

Row Echelon Form (REF)

A matrix is in reduced row echelon form if:

• The zero rows (if any) are at the bottom of the matrix.
• The leading (first non-zero) entry in each non-zero row is to the right of the preceding row’s leading entry.
• Every entry that comes after a leading entry is zero.

Example:

Reduced Row Echelon Form (RREF)

A matrix is in reduced row echelon form if:

• It meets every requirement of the row echelon form.
• Every non-zero row has a leading entry of 1 (called a leading 1).
• Each leading 1 represents the only non-zero value in its column.

Example:

Here’s an example of converting a matrix into Row Echelon Form (REF) and Reduced Row Echelon Form (RREF) using Gaussian and Gauss-Jordan elimination, respectively.

Problem:

Step1: Transform to row echelon form

To add zeros below the pivot places, use row operations.

1. Divide the first row by 2 (make the leading entry 1):

2. Eliminate the first column entries below the pivot (1 in R1​[1,1])

Now, the matrix is in Row Echelon Form (REF).

Step 2: Transform to Reduced Row Echelon Form (RREF)

Use row operations to set the pivot entries to 1 and remove any non-zero items from the pivot columns.

1. Make the pivot in the second row 1 (already done) and eliminate the second column entries above and below it:

2. Make the pivot in the third row 1 by dividing R3 by 5:

3. Remove the third column entries above the pivot in R 3 [3, 3].

Now, the matrix is in Reduced Row Echelon Form (RREF).

Final forms:

Row Echelon Form (REF):

Reduced Row Echelon Form (RREF):