row echelon form

Therefore only row echelon forms are considered in the remainder of this article.

However, every matrix has a unique reduced row echelon form.

To find a basis, we reduce A to row echelon form:

If this is the case, then matrix is said to be in row echelon form.

This is equal to the number of pivots in the reduced row echelon form.

Use elementary row operations to put A into row echelon form.

This is an example of a matrix in reduced row echelon form:

It produces the following row echelon form of 'A':

For example, the following matrix is in row echelon form, and its leading coefficients are shown in red.

They are also used in Gauss-Jordan elimination to further reduce the matrix to reduced row echelon form.

This matrix is then modified using elementary row operations until it reaches reduced row echelon form.

The reduced row echelon form of a matrix may be computed by Gauss-Jordan elimination.

The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form.

Two matrices in reduced row echelon form have the same row space if and only if they are equal.

Note that the independent columns of the reduced row echelon form are precisely the columns with pivots.

If we instead put the matrix A into reduced row echelon form, then the resulting basis for the row space is uniquely determined.

For example, row echelon form and Jordan normal form are canonical forms for matrices.

If the final column of the reduced row echelon form contains a pivot, then the input vector v does not lie in S.

With the modern computers, Gaussian elimination is not always the fastest algorithm to compute the row echelon form of matrix.

The final matrix (in reduced row echelon form) has two non-zero rows and thus the rank of matrix A is 2.

A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions.

Using these operations a matrix can always be transformed into an upper triangular matrix, and in fact one that is in row echelon form.

Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss-Jordan elimination.

Some authors use the term Gaussian elimination to refer to the process until it has reached its upper triangular, or (non-reduced) row echelon form.

By performing row operations, one can check that the reduced row echelon form of the this augmented matrix is: