by Marco Taboga, PhD
A matrix is said to be in reduced row echelon form when it is in row echelon form and its basic columns are vectors of the standard basis (i.e., vectors having one entry equal to 1 and all the other entries equal to 0).
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When the coefficient matrix of a linear system is in reduced row echelon form, it is straightforward to derive the solutions of the system from the coefficient matrix and the vector of constants.
Prerequisites
In order to understand this lecture, you should first read the lecture on the Row echelon form.
In particular, remember that a matrix is in row echelon form if and only if:
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all its non-zero rows have an entry, called pivot, that is non-zero and has only zero entries below it and to its left;
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zero-rows (if there are any) are below the non-zero rows.
When a column of a matrix in row echelon form contains a pivot, it is called a basic column. When it does not contain a pivot, we say that it is a non-basic column.
Definition of reduced row echelon form
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A precise definition of reduced row echelon form follows.
We show some matrices in reduced row echelon form in the following examples.
How to solve a system in reduced echelon form
Consider a linear system where is a matrix of coefficients, is an vector of unknowns, and is a vector of constants.
The system is said to be in reduced row echelon form if the matrix is in reduced row echelon form.
As explained in the lecture on Matrix multiplication and linear combinations, the product can be written as a linear combination of the columns of : where the coefficients of the combination are the unknowns .
If an unknown multiplies a basic column, it is called a basic variable. Otherwise, if it corresponds to a non-basic column, it is called a non-basic variable.
Since the reduced echelon form is a special case of the echelon form, the conditions for the existence of a solution of a system in the latter form apply, and we can use the back-substitution algorithm to solve the system.
Remember how the back-substitution algorithm works:
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if there are basic columns, we choose values arbitrarily for the non-basic variables (i.e., for the unknowns corresponding to the non-basic columns);
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for , if the -th row is non-zero and is the basic variable corresponding to the pivot of the -th row, we set
If is in reduced row echelon form, then , so that equation 1 becomesFurthermore, the coefficients in equation 2 are equal to when is the index of a basic column.
How to transform a system to reduced row echelon form
The standard algorithm used to transform a system into an equivalent system in reduced row echelon form is called Gauss Jordan elimination.
Solved exercises
Below you can find some exercises with explained solutions.
Exercise 1
Determine whether the matrixis in reduced row echelon form.
Exercise 2
Determine whether the matrixis in reduced row echelon form.
How to cite
Please cite as:
Taboga, Marco (2021). “Reduced row echelon form”, Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/reduced-row-echelon-form.
Source: https://t-tees.com
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