## Introduction to RREF Calculator:

The rref calculator is a powerful tool that is used to **convert the matrix** into a reduced echelon form in a run of time. It helps you to determine the large matrix to get the solution of a given system of equations in linear algebra.

When you deal with large matrices it would be difficult to solve the complex problem manually, as when you do any error in row operation you cannot achieve the desired result. That is why we introduce the reduced row echelon form calculator that does all your work, just enter the values in the tool and get the solution to your problem.

## What is RREF?

Rref is the abbreviation of **reducing row echelon form** in linear algebra. This method is used to convert the given matrix into reduced echelon form in which all the diagonal elements have entity 1 and all other elements except the diagonal are zero in the given matrix.

It uses the Gauss elimination or Gauss-Jordan elimination method to reduce the matrix so that you get the variable values from the system of linear equations.

## How to Calculate RREF?

To **calculate the reduced row echelon** form (RREF) of a matrix, you need to know about the row operation. There are some steps followed by the row reduced echelon form calculator in order to reduce the given matrix into an identity matrix.

**Step 1**:

Convert the given system of the equation into an augmented matrix.

**Step 2**:

After conversion into matrix, use row operation to make the first entity 1 in the left-right on the diagonal

**Step 3**:

In row operation, you can use the basic mathematical method like addition, subtraction, multiplication, and division to convert all the entries in the matrix is zero except the diagonal matrix, which should have the entity 1 in each row.

**Step 4**:

By following these steps systematically, you can transform any matrix into its RREF and get the variable values in solution from the given system of linear equations easily.

## Practical Example of RREF

An **example of RREF** with the solution is given to understand the calculation process of the rref calculator.

**Example**:

Find the rref matrix of the system of equations.

$$ 2x + 5y + 4z \;=\; 12 $$

$$ 3x + y - 4z \;=\; 1 $$

$$ x - 2y - 3z \;=\; 0 $$

**Solution**:

First, convert the system of equations into matrix form,

$$ \biggr|\begin{matrix} 2 & 5 & 4 & 12 \\ 3 & 1 & -4 & 1 \\ 1 & -2 & -3 & 0 \\ \end{matrix} \biggr| $$

Now we use the row operation to achieve the reduced echelon form of this matrix. Replace R1 with R3 because R3 has the leading entity 1.

$$ -3(R_1) + R_2 \rightarrow R_2 $$

$$ \biggr| \begin{matrix} 1 & -2 & -3 & 0 \\ 3 & 1 & -4 & 1 \\ 2 & 5 & 4 & 12 \\ \end{matrix} \biggr| $$

\begin{array}{rrrr} -3 & 6 & 9 & 0 \\ 3 & 1 & -4 & 1 \\ \hline 0 & 7 & 5 & 1 \\ \end{array}

Then multiply -3 with R1 and add it with R2

$$ -2(R_1) + R_3 \rightarrow R_3 $$

$$ \biggr| \begin{matrix} 1 & -2 & -3 & 0 \\ 0 & 7 & 5 & 1 \\ 2 & 5 & 4 & 12 \\ \end{matrix} \biggr| $$

\begin{array}{rrrr} -2 & 4 & 6 & 0 \\ 2 & 5 & 4 & 12 \\ \hline 0 & 9 & 10 & 12 \\ \end{array}

Then multiply -2 with R3 and add it with R3

$$ 1/7(R_2) \rightarrow R_2 $$

$$ \biggr| \begin{matrix} 1 & -2 & -3 & 0 \\ 0 & 7 & 5 & 1 \\ 0 & 9 & 10 & 12 \\ \end{matrix} \biggr| $$

After dividing R2 by 1/7

$$ 2(R_2) + R_1 \rightarrow R_1 $$

$$ \biggr| \begin{matrix} 1 & -2 & -3 & 0 \\ 0 & 1 & 5/7 & 1/7 \\ 0 & 9 & 10 & 12 \\ \end{matrix} \biggr| $$

\begin{array}{rrrr} 0 & 2 & 10/7 & 2/7 \\ 1 & -2 & -3 & 0 \\ \hline 1 & 0 & -11/7 & 2/7 \\ \end{array}

Multiply -9 with R2 and then add with R3.

$$ -9(R_2) + R_3 \rightarrow R_3 $$

$$ \biggr| \begin{matrix} 1 & 0 & -11/7 & 2/7 \\ 0 & 1 & 5/7 & 1/7 \\ 0 & 9 & 10 & 12 \\ \end{matrix} \biggr| $$

\begin{array}{rrrr} 0 & -9 & 45/7 & -9/7 \\ 0 & 9 & 10 & 12 \\ \hline 0 & 0 & 25/7 & 75/7 \\ \end{array}

To make 1 in the last diagonal element divide R3 with 7/25.

$$ \biggr| \begin{matrix} 1 & 0 & -11/7 & 2/7 \\ 0 & 1 & 5/7 & 1/7 \\ 0 & 0 & 25/7 & 75/7 \\ \end{matrix} \biggr| \; \; \; \; \; 7/25(R_3) \rightarrow R_3 $$

After getting entity 1 in R3, 11/7 with R3 and then add with R1 so that you can easily make it zero.

$$ 11/7(R_3) + R_1 \rightarrow R_1 $$

$$ \biggr| \begin{matrix} 1 & 0 & -11/7 & 2/7 \\ 0 & 1 & 5/7 & 1/7 \\ 0 & 0 & 1 & 3 \\ \end{matrix} \biggr| $$

\begin{array}{rrrr} 0 & 0 & 11/7 & 33/7 \\ 1 & 0 & -11/7 & 2/7 \\ \hline 1 & 0 & 0 & 5 \\ \end{array}

Multiply R3 with -5/7 and then add with R2 so that it gives the zero after the leading entity in the diagonal.

$$ -5/7(R_3) + R_2 \rightarrow R_2 $$

$$ \biggr| \begin{matrix} 1 & 0 & 0 & 5 \\ 0 & 1 & 5/7 & 1/7 \\ 0 & 0 & 1 & 3 \\ \end{matrix} \biggr| $$

\begin{array}{rrrr} 0 & 0 & -5/7 & -15/7 \\ 0 & 1 & 5/7 & 1/7 \\ \hline 0 & 1 & 0 & -2 \\ \end{array}

After row operation the given matrix changes into a reduced echelon form where all values in the matrix (upper and lower triangle are zero) except the leading entity on the diagonal.

$$ \biggr| \begin{matrix} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 3 \\ \end{matrix} \biggr| $$

Again change the matrix into equation form to get the variable values (x,y,z) such as:

$$ 1x + 0y + 0z \;=\; 5 $$

$$ 0x + 1y + 0z \;=\; -2 $$

$$ 0x + 0y + 1z \;=\; 3 $$

$$ 1x 0y + 0z \;=\; 5 \rightarrow x \;=\; 5 $$

$$ 0x + 1y + 0z \;=\; -2 \rightarrow y \;=\; -2 $$

$$ 0x + 0y + 1z \;=\; 3 \rightarrow z \;=\; 3 $$

Thus the variable value of given system of linear equation (x,y,z) is (5,-2,3).

## How to Use RREF Calculator

The reduced row echelon form calculator has a simple interface, that enables you to use it to evaluate the reduced row echelon form from the given system of equation questions. Before adding the input for the solutions, you must follow some simple steps. These steps are:

**Select the order**of the matrix from the list for the Rref solution.- Enter the matrix element as per your matrix order in the input box.
- Review your input value for the reduced row echelon form solution before hitting the calculate button of row reduction calculator to start the calculation process.
- Click on the “Calculate” button to get the desired result of your given system of linear equation problems.
- If you want to try out the row reduced echelon form calculator to check its accuracy in the solution then you should try the load example.
- Click on the “Recalculate” button to get a new page for solving more linear system questions.

## Final Result of Reduced Row Echelon Form Calculator

The rref calculator gives you the **solution** to a given problem when you add the input to it. It provides you with solutions of linear systems. It may contain as:

**Result Option**

You can click on the result option and the rref matrix calculator provides you with a solution in the form of a reduced row echelon form.

**Possible Step**

When you click on the possible steps option it provides you with the solution from the given linear equation using the rref method.

## Benefits of Row Reduced Echelon Form Calculator

The reduced row echelon calculator gives you millions of benefits, you get whenever you use it to calculate a system of linear equations and to get its solution. These benefits are:

- Our reduced row echelon form calculator
**saves the time and effort**that you consume in solving complex linear equation questions immediately. - It is a free tool that provides you solution in the form of reduced row echelon form from the given system without spending.
- The row reduction calculator is an adaptive tool that enables you to find the rref for complex or large matrix problems from the given system of equations.
- You can use this rref matrix calculator for practice so that you get familiar with the rref concept.
- The rref calculator is a trustworthy tool that provides you with accurate solutions as per your input system of equations to calculate its solution using rref process.