# RREF Calculator

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The RREF calculator simplifies and organizes a system of linear equations represented in matrix form and transforms them into a reduced row echelon form. This is done by applying a series of row operations such as swapping rows, multiplying rows by non-zero constants, and adding multiples of one row to another.

## How to use this reduced row echelon form calculator?

Calculating the Reduced Row Echelon Form (RREF) using our RREF calculator is quite simple, It only requires a few inputs to perform, including:

• First, Set the size of the matrix for which you are going to calculate
• Then, Enter the values of the matrix in the required field
• Finally, tap on the Calculate button

The RREF calculator will quickly process the information and provide you with the reduced echelon form of the matrix along with step-by-step solutions.

## What is reduced row echelon form (RREF)?

The reduced row echelon form (RREF) is a standardized and simplified representation of a matrix achieved through a series of row operations being applied.

A matrix is in reduced row echelon form when it meets three conditions:

• It's already in a row echelon form
• All its pivots (the first non-zero entry in each row) are 1
• The pivots are the only non-zero numbers in their respective columns

## How do I find the RREF of a matrix?

Let's go through an example of finding the RREF of a matrix for better understanding, Here are the steps:

### Example:

Consider the following matrix:

$A = \begin{bmatrix} 2 & 1 & 3 \\ 1 & 2 & 1 \\ 3 & 3 & 5 \end{bmatrix}$

Step 1: Divide row 1 by 2:

• $\begin{bmatrix}1 & 0.5 & 1.5 \\1 & 2 & 1 \\3 & 3 & 5\end{bmatrix}$

Step 2: Subtract row 1 multiplied by 1 from row R2:

• $\begin{bmatrix}1 & 0.5 & 1.5 \\0 & 1.5 & -0.5 \\3 & 3 & 5\end{bmatrix}$

Step 3: Subtract row 2 multiplied by 3 from row R3:

• $\begin{bmatrix}1 & 0.5 & 1.5 \\0 & 1.5 & -0.5 \\0 & -0.5 & 0.5\end{bmatrix}$

Step 4: Multiply row 2 by 2/3:

• $\begin{bmatrix}1 & 0.5 & 1.5 \\0 & 1 & -1/3 \\0 & -0.5 & 0.5\end{bmatrix}$

Step 5: Subtract row 0 multiplied by 1/2 from row R1:

• $\begin{bmatrix}1 & 0 & 0.5 \\0 & 1 & -1/3 \\0 & -0.5 & 0.5\end{bmatrix}$

Step 6: Subtract row 2 multiplied by 1/2 from row R3:

• $\begin{bmatrix}1 & 0 & 0.5 \\0 & 1 & -1/3 \\0 & 0 & 1\end{bmatrix}$

Step 7: Multiply row 3 by 3/2:

• $\begin{bmatrix}1 & 0 & 0.5 \\0 & 1 & -1/3 \\0 & 0 & 1\end{bmatrix}$

Step 8: Subtract row 0 multiplied by 5/3 from row R1:

• $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & -1/3 \\0 & 0 & 1\end{bmatrix}$

Step 9: Subtract row 1 multiplied by -1/3 from row R2:

• $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$

So, this is the final reduced row echelon form of the given matrix. Now that you have gone through the process, we hope you have gained a clear understanding of how to determine the reduced row echelon form (RREF) of any matrix using the RREF calculator provided by Calculatored.

### Alan Walker

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.