Row Echelon Form

Definition

The Row Echelon Form (REF) is a specific form of a matrix obtained through elementary row operations. The leftmost non-zero entry of each row, known as the leading entry or pivot, is 1 and each pivot is to the right of the pivot in the row above. All rows that contain only zeros, if any, are at the bottom of the matrix matrix.

Properties

A matrix is in row echelon form if it satisfies the following conditions:

All rows that contain only zeros are at the bottom.
The leading entry of each non-zero row is 1.
The leading entry of a non-zero row is always strictly to the right of the leading entry of the row above.

Elementary Row Operations

Elementary row operations are used to transform a matrix into row echelon form. There are three types of elementary row operations:

Swapping two rows
Multiplying a row by a non-zero scalar
Adding a multiple of one row to another row

Gaussian Elimination

Gaussian elimination is a method used to put a matrix into row echelon form. It involves using the elementary row operations to create a matrix where the leading entry of each row is 1 and each pivot is to the right of the pivot in the row above.

Reduced Row Echelon Form

The Reduced Row Echelon Form (RREF) is a form of matrix that is in row echelon form and also satisfies the following additional conditions:

The pivot of each row is the only non-zero entry in its column.
The leading entry in each non-zero row is 1.

Applications

Row echelon form and reduced row echelon form are used in various fields of mathematics, including linear algebra, calculus, and differential equations. They are particularly useful for solving systems of linear equations and finding the rank of a matrix.