RREF Calculator with steps | Matrix Reduced Row Echelon Form Calculator

RREF Calculator with steps

Size of the matrix: x

Knowing how to find Reduced Row Echelon Form (RREF) of a matrix is of great help in solving systems of linear equations. For this reason, we put at your hands this RREF calculator with steps, which allows you to quickly and easily reduce a matrix to row echelon form.

To use this calculator you must follow these simple steps:

  1. Enter the dimensions of the matrix you want to reduce.
  2. Enter the matrix in the fields intended for it.
  3. Press the “Calculate RREF” button, doing so will automatically display a box with the detailed step-by-step solution.

Below you will find a summary of the most important theoretical concepts related to how to do reduced row echelon form.

What is Reduced Row Echelon Form (RREF) of a matrix?

The Reduced Row Echelon Form of a matrix A is another matrix H that satisfies the following properties:

  • It has rows composed entirely of zeros (null rows), these are grouped at the bottom of the matrix.
  • The pivot (first non-null element) of each non-null row is 1.
  • The pivot of each nonzero row is to the right of that of the previous row.
  • Items that appear in the same column as the pivot of a row are all zero.

Here is an example of the reduced row echelon form of a matrix:

Every matrix has a single row-reduced echelon form, regardless of how you perform operations on the rows.

The importance of matrices in reduced row echelon form comes from the following theorem:

Each matrix can be transformed into reduced row echelon form by a sequence of elementary row operations.

Reduced Row Echelon Form is useful because it provides a standard form for writing matrices and systems of linear equations that makes it easy to solve them. For example, if a matrix is in Reduced Row Echelon Form, you can easily find the solutions to the corresponding system of linear equations by reading the values of the variables from the matrix.

How to find Reduced Row Echelon Form

To carry out this process, it is necessary to carry out a succession of elementary row transformations, which are:

  • Type I: Swap the position of two rows.
  • Type II: Multiply all elements of a row by a non-zero scalar.
  • Type III: Add to one row another multiplied by a scalar.
  1. We can start by getting a pivot 1 in the first row. If in the first column there is some element that is not null, we will get this pivot in position (1,1). If all the elements in the first column are zero, we move on to trying the second (position (1,2)) and so on. Depending on the case, there are several ways to get this pivot 1, (of course we can get it on any row and then exchange to take it to the first one).
  2. In each of the remaining rows, the element located below the pivot becomes 0 by adding the first multiplied by the convenient scalar (type III transformation). Once this is done, the matrix will have one of the following forms:
  1. We have to repeat the process (steps 1 and 2) for the following rows, until there are no more or all the elements of the remaining rows are zero.
  2. Finally, with the pivot 1 of each non-null row, the corresponding term of all the previous ones is made 0, so that the resulting matrix will be in the rows reduced echelon form.

Reduced Row Echelon Form examples and solutions

Here are some examples that will help you better understand what was explained above. These examples have been created using the RREF Calculator with steps.

Example 01: Find the reduced row echelon form of this matrix 2×2

matrix 2x2 example 01


matrix reduced row echelon form example 01 - solution

Example 02: Find the reduced row echelon form of this matrix 3×3

matrix reduced row echelon form example 02


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