Introduction
This RREF calculator turns any matrix into reduced row echelon form or row echelon form (REF) using Gauss-Jordan elimination. Just type your numbers into the grid, press Calculate, and get the answer right away. The tool shows every row operation step by step so you can follow along and learn how the process works.
It works with both coefficient matrices and augmented matrices. You can enter whole numbers, decimals, or fractions like 2/3. The calculator finds the rank, pivot columns, free variables, and nullity of your matrix. For augmented matrices, it also tells you whether the system of linear equations has a unique solution, infinitely many solutions, or no solution at all.
All answers are computed using exact fractions, so you never lose accuracy to rounding. You can also switch to decimal mode if you prefer. This free online RREF calculator gives you clear, detailed results.
How to Use Our RREF Calculator
Enter your matrix values and settings below. The calculator will show the reduced row echelon form (RREF) or row echelon form (REF) of your matrix, along with step-by-step row operations, rank, pivot positions, and solution analysis.
Set your matrix size by typing the number of rows and columns (1 to 10 each) into the Rows and Columns fields, then click "Generate Matrix" to build the grid.
Choose the matrix type. Pick "Augmented" if your matrix includes a constants column on the right side (used for solving systems of equations). Pick "Coefficient" if your matrix has no constants column.
Select the output form. Choose "RREF" for the fully reduced row echelon form, or choose "REF" for the standard row echelon form.
Pick how values are shown. Use the "Display Values As" dropdown to see results as exact fractions or as decimals rounded to four decimal places.
Enter your numbers into the grid. Type an integer, decimal, or fraction (like 2/3 or -1.5) into each cell. Use Tab, Shift+Tab, or the arrow keys to move between cells. Press Enter to calculate right away.
Or paste a full matrix. Click "Paste a Matrix" to open a text box. Type or paste your matrix with values separated by spaces, commas, or tabs, and rows on separate lines. Then click "Load from Text" to fill the grid automatically.
Try a quick preset by clicking any example button (like "Unique Solution" or "No Solution") to load a sample matrix and see results instantly.
Click "Calculate" to run the row reduction. The calculator will display your original matrix, every row operation step by step, the final RREF or REF matrix with pivot positions highlighted, the rank and nullity, and a full solution analysis for augmented matrices.
What Is Reduced Row Echelon Form (RREF)?
Reduced row echelon form, or RREF, is a way to simplify a matrix so it is as easy to read as possible. A matrix is a grid of numbers arranged in rows and columns. When you put a matrix into RREF, you use simple row operations to reshape it until each leading number in a row is 1, every other number in that leading column is 0, and the leading 1s step to the right as you move down the rows.
What Are Row Operations?
To reach RREF, you only need three moves, called elementary row operations:
- Swap: Switch two rows with each other.
- Scale: Multiply every number in a row by the same non-zero value.
- Replace: Add or subtract a multiple of one row from another row.
These operations change the look of the matrix but never change the solution of the system it represents.
REF vs. RREF
Row echelon form (REF) is a partially simplified version. It only requires zeros below each leading entry. Reduced row echelon form (RREF) goes further and also clears all numbers above each leading entry, making each leading entry equal to 1. RREF gives you the cleanest, most direct answer.
Why Does RREF Matter?
RREF is one of the most important tools in linear algebra. It lets you solve systems of linear equations, find the rank of a matrix, identify pivot and free variables, and determine whether a system has one solution, infinitely many solutions, or no solution at all. It is the standard method taught in algebra and college math courses for working with matrices by hand or with a calculator.
RREF is also closely related to other matrix operations. For example, knowing the rank helps you determine whether an inverse matrix exists, and the pivot structure connects directly to the determinant of a square matrix.
Augmented vs. Coefficient Matrices
An augmented matrix includes the constants from the right side of a system of equations, separated by a vertical line. A coefficient matrix holds only the numbers in front of the variables. Use the augmented form when you want to solve a system. Use the coefficient form when you need the rank or other properties of the matrix alone.
Understanding the Results
After row reduction, the rank tells you how many independent equations exist. Pivot columns point to the leading variables, and the remaining columns point to free variables. If there are no free variables in an augmented system, the solution is unique. If free variables exist, there are infinitely many solutions. If a row reduces to something like 0 = 5, the system has no solution and is called inconsistent.