# Reduced Row Echelon Form (RREF) Calculator

## Find reduced row echelon form step by step

The calculator will find the row echelon form (simple or reduced – RREF) of the given (augmented if needed) matrix, with steps shown.

Related calculators: Gauss-Jordan Elimination Calculator, Matrix Inverse Calculator

The RREF Calculator is an online resource designed to convert matrices into RREF. This calculator assists you in solving systems of linear equations by putting a matrix into a row echelon form. It also helps us understand the underlying processes behind these computations.

## How to Use the Reduced Row Echelon Form Calculator?

### Input

Provide the elements of your matrix in the specified fields.

### Calculation

Once the matrix is entered, click on the "Calculate" button.

### Result

The calculator will immediately process the data and present the Reduced Row Echelon Form of your matrix.

## What Is Reduced Row Echelon Form?

The Reduced Row Echelon Form (RREF) is an important concept in linear algebra. When a matrix is in RREF, it allows for a straightforward interpretation of the solution of the system of linear equations.

Here's a more detailed explanation using an example. Consider the following system of three linear equations:

$$\begin{cases}x+2y+z=9 \\2x+4y+z=18\\3x+5y+z=24\end{cases}$$We can represent this system in matrix form as follows:

$$\left[\begin{array}{cccc}1&2&1&9\\2&4&1&18\\3&5&1&24\end{array}\right]$$The goal is to transform this matrix into its RREF using Gauss-Jordan elimination. The RREF of a matrix must meet the following conditions:

- If a row has non-zero entries, then the leftmost non-zero entry is a $$$1$$$, also called the leading $$$1$$$.
- Any column that contains a leading $$$1$$$ has all other entries as $$$0$$$.
- The leading $$$1$$$ in any row is to the right of the leading $$$1$$$ in the row above.
- All rows with all zero entries are at the bottom.

Applying elementary row operations (EROs) to the above matrix, we subtract the first row multiplied by $$$2$$$ from the second row and multiplied by $$$3$$$ from the third row to eliminate the leading entries in the second and third rows.

After the EROs, the matrix becomes:

$$\left[\begin{array}{cccc}1&2&1&9\\0&0&-1&0\\0&-1&-2&-3\end{array}\right]$$Swap the second and third rows:

$$\left[\begin{array}{cccc}1&2&1&9\\0&-1&-2&-3\\0&0&-1&0\end{array}\right]$$Multiply the second row by $$$-1$$$:

$$\left[\begin{array}{cccc}1&2&1&9\\0&1&2&3\\0&0&-1&0\end{array}\right]$$Subtract the second row multiplied by $$$2$$$ from the first row:

$$\left[\begin{array}{cccc}1&0&-3&3\\0&1&2&3\\0&0&-1&0\end{array}\right]$$Multiply the third row by $$$-1$$$:

$$\left[\begin{array}{cccc}1&0&-3&3\\0&1&2&3\\0&0&1&0\end{array}\right]$$Add the third row multiplied by $$$3$$$ to the first row and multiplied by $$$-2$$$ to the second row:

$$\left[\begin{array}{cccc}1&0&0&3\\0&1&0&3\\0&0&1&0\end{array}\right]$$This RREF matrix corresponds to the solution $$$x=3$$$, $$$y=3$$$, and $$$z=0$$$ for our original system of equations.

## Why Choose Our Reduced Row Echelon Form Calculator?

### User-Friendly Interface

The calculator is designed to be simple and intuitive, targeting users with different levels of mathematical knowledge.

### Fast and Accurate

Our calculator delivers instantaneous and precise results, which can significantly save your time and reduce potential calculation errors.

### Handles Complex Calculations

It can handle matrices of different dimensions, allowing for different applications, from simple to more complex systems of equations.

### Educational Value

It not only delivers the solution but also helps you understand the process behind Gauss-Jordan elimination, making it a valuable learning tool.

### FAQ

#### Can the RREF Calculator handle large matrices?

Absolutely. The RREF Calculator is capable of managing matrices of different dimensions.

#### How does the RREF Calculator work?

The RREF Calculator uses a mathematical procedure known as Gauss-Jordan elimination to reduce matrices to their row echelon form. This method involves a sequence of row operations to transform the matrix.

#### What are the advantages of using the RREF Calculator?

The RREF Calculator provides accurate and quick results, simplifying matrix transformations. It's an ideal tool for students, educators, and professionals needing to handle complex mathematical operations.

#### What is a Reduced Row Echelon Form (RREF)?

The Reduced Row Echelon Form (RREF) is a special form of a matrix. It helps simplify the process of solving systems of linear equations. A matrix in RREF has ones as leading entries in each row, with all other entries in the same column as zeros. All rows of zeros are at the bottom of the matrix.