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.