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Math Calculators ▶ RREF Calculator

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**Table of Content**

Use this handy rref calculator that helps you to determine the reduced row echelon form of any matrix by row operations being applied.

So stay connected to learn the technique of matrix reduction and how this reduced row echelon form calculator will assist you to amplify your speed of calculations.

A matrix is said to be in either echelon or reduced echelon form if it satisfies the following set of conditions:

- It is already in echelon form
- All of its pivots are equal to 1 considering that the pivots are the only elements that are considered as non-zero ones

Let’s resolve an example to clarify your concept of both echelon and reduced echelon form. Stay focused!

**Example # 01:**

Find the reduced echelon form of the matrix given below:

$$ \begin{bmatrix} 2 & 3 & 8 \\ 14 & 8 & -7 \\ 7 & -3 & 1 \\\end{bmatrix} $$

**Solution:**

As the given matrix is:

$$ \begin{bmatrix} 2 & 3 & 8 \\ 14 & 8 & -7 \\ 7 & -3 & 1 \\\end{bmatrix} $$

Determination of Reduced Echelon Form:

**Step # 01:**

Divide first row by 2:

$$ \begin{bmatrix} 1&\frac{3}{2}&4 \\14&8&-7 \\ 7&-3&1\\\end{bmatrix} $$

**Step # 02:**

Multiply first row by 14 and subtract it from first row:

$$ \begin{bmatrix}1&\frac{3}{2}&4 \\ 0&-13&-63 \\ 7&-3&1 \\\end{bmatrix} $$

**Step # 03:**

Multiply second row by 7 and minus it from the third row:

$$ \begin{bmatrix}1&\frac{3}{2}&4 \\0&-13&-63 \\0&\frac{-27}{2}&-27 \\\end{bmatrix} $$

**Step # 04:**

Go for dividing the second row by -13:

$$ \begin{bmatrix}1&\frac{3}{2}&4 \\0&1&\frac{24230769231}{5000000000} \\0&\frac{-27}{2}&-27 \\\end{bmatrix} $$

We understand that these suppositions in calculations may be confusing but do not worry as this rref calculator

**Step # 04:**

Now it’s time to multiply zeroth row by the number 3/2 and subtract it from first row:

$$ \begin{bmatrix}1&0&\frac{-8173076923}{2500000000} \\0&1&\frac{24230769231}{5000000000} \\0&\frac{-27}{2}&-27 \\\end{bmatrix} $$

**Step # 05:**

Multiply second row by -27/2 and minus it from the third row:

$$ \begin{bmatrix}1&0&\frac{-8173076923}{2500000000} \\0&1&\frac{24230769231}{5000000000} \\0&0&\frac{19211538462}{500000000} \\\end{bmatrix} $$

**Step # 06:**

Move on multiplying third row by 500000000/19211538462:

$$ \begin{bmatrix}1&0&\frac{-8173076923}{2500000000} \\0&1&\frac{24230769231}{5000000000} \\0&0&1 \\\end{bmatrix} $$

**Step # 07:**

Multiply zeroth row by -8173076923/2500000000 and subtract it from first row:

$$ \begin{bmatrix}1&0&0 \\0&1&\frac{24230769231}{5000000000} \\0&0&1\\\end{bmatrix} $$

**Step # 07:**

Get the product of first row with 24230769231/5000000000 and subtract the result from second row as below:

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

As all principal diagonal elements have zero on above and below them, the form generated is considered a reduced echelon.

This row reduced echelon form calculator will take a couple of moments to generate the row echelon form of any matrix.

**Input:**

- First of all, set up the order of the matrix by fixing the number of rows and columns from first and second lists, respectively
- After you do that, tap “Set Matrices” to et the proper layout of the final matrix
- Now enter the entities of the matrix in in the designated fields of the row echelon calculator
- At last, hit the calculate button

**Output:**

- The reduced echelon form calculator will reduce the matrix in reduced echelon form
- Shows all row operations involved in reducing the given matrix

From the source of wikipedia: Row echelon form, Reduced row echelon form, Transformation, Systems of linear equations, Gaussian elimination, Applications

From the source of khan academy: Matrix row operations