RREF Calculator

Input a matrix to instantly find the accurate Reduced Row Echelon Form (RREF) via row operations.

Size of the matrix:

Reduced

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RREF Calculator

Our RREF calculator computes the Reduced Row Echelon Form of any matrix by performing row operations step by step. It is designed for students, teachers, and professionals to simplify linear algebra tasks, making matrix calculations faster and more precise.

What Is Reduced Row Echelon Form?

A matrix is in reduced row echelon form (RREF) if it satisfies all of the following criteria:

The matrix is in Row Echelon Form (REF)
Every leading entry (pivot) in a non-zero row is 1
Each pivot is the only non-zero entry in its column

For instance, this matrix is in RREF:

$ \begin{bmatrix} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 3 \end{bmatrix} $

Steps to Convert a Matrix to RREF

Locate the leftmost column that contains a non-zero element.
If necessary, create a leading 1 at the top of that column.
Use row operations to make all other entries in that column zero.
Move to the next non-zero column below the previous pivot and repeat.
Continue until each non-zero row contains a leading 1.
Eliminate all entries above each pivot so that every pivot is the only non-zero value in its column.
Verify that the matrix meets all RREF conditions.

🔎 These steps mirror what the RREF calculator performs automatically. Understanding this process also helps when computing eigenvalues. Check out our Eigenvalues and Eigenvectors Calculator for guided solutions.

Example:

Find the RREF of the matrix:

$$ \begin{bmatrix} 3 & 5 \\ 7 & 9 \end{bmatrix} $$

Solution:

Step 1: Divide the first row by 3 → $ R_1 = R_1/3 $

$ \begin{bmatrix} 1 & 5/3 \\ 7 & 9 \end{bmatrix} $

Step 2: Eliminate the entry below the pivot → $ R_2 = R_2 - 7R_1 $

$ \begin{bmatrix} 1 & 5/3 \\ 0 & -8/3 \end{bmatrix} $

Step 3: Scale the second row → $ R_2 = (-3/8)R_2 $

$ \begin{bmatrix} 1 & 5/3 \\ 0 & 1 \end{bmatrix} $

Step 4: Eliminate the entry above the second pivot → $ R_1 = R_1 - (5/3)R_2 $

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

How to Use the RREF Calculator

Choose Matrix Size: Select the number of rows and columns for your matrix.
Input Matrix Values: Enter each element into the corresponding input field.
Calculate: Click "Calculate" to obtain the RREF along with step-by-step row operations.