What is Reduced Row Echelon Form?
Reduced Row Echelon Form is a form of matrix used to find the matrix's rank and solve a system of linear equations. In other words, a matrix is in reduced row echelon form only if it is row echelon form and its pivots are the only non-zero entries of the basic columns and are equal to 1.
The Gauss-Jordan elimination process usually computes the reduced row echelon form. Unlike the row echelon form, the reduced row echelon form does not depend on the algorithm to compute it. Even though the row echelon form is not unique, all the row echelon forms and the reduced row echelon form have the same number of zero rows, and the pivots are located in the same indices.
What are the Benefits of Reduced Row Echelon Form?
- Reduced Row Echelon Form offers a singular basis for the row space to compare two matrices or to determine if the two lists of vectors span the same subspace.
- You can easily see the null space of a matrix from the Reduced Row Echelon Form.
- In the Reduced Row Echelon Form, you can easily see the solution to a system of linear equations.
How Do You Calculate the Reduced Row Echelon Form?
Reduced Row Echelon Form uses Gauss and Gauss-Jordan elimination which uses matrix-row reduction, which in turn relies on the elementary row operations, primarily -
- You can switch any two equations.
- You can multiply any equation by a non-zero constant number.
- You can add any non-zero multiple to any equation.
The easiest way to calculate Reduced Row Echelon Form is to use our calculator - RREF Calculator. All you need to do is enter the matrices separated by spaces and new lines, and you will get the appropriate result.
FAQs
Can every matrix be put in reduced row echelon form?
Any matrix can be transformed into a reduced row echelon form using a technique known as Gauss-Jordan elimination, which is usually used to solve problems involving linear equations.
Can reduced row echelon form be inconsistent?
The system is consistent if you encounter any inconsistent row during row-reduction.
Why do you need a reduced row echelon form?
The reduced row echelon form is important as it lets you analyze if the linear equation system corresponding to the augmented matrix is solvable.
How many solutions are there in a reduced row echelon form?
The solution depends on the variables and the row -
- If there is no row in reduced row echelon form, there might be one or infinite solutions.
- If there is a solution, there is a free variable, which means it has infinite solutions.
- There is only one solution if there is a solution and no free variable.