A matrix is in row echelon form (ref) when it satisfies the following conditions.
A matrix is in reduced row echelon form (rref) when it satisfies the following conditions.
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.
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.
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.
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1. What is echelon form in linear algebra? 
2. How is echelon form different from reduced echelon form? 
3. What are the benefits of using echelon form in linear algebra? 
4. How can a matrix be transformed into echelon form? 
5. Can every matrix be transformed into echelon form? 

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