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RREF Calculator

RREF Calculator

Convert matrices to Reduced Row Echelon Form with detailed step-by-step row operations. Perfect for solving linear systems, finding matrix rank, and understanding row operations.

100% FreeStep-by-step SolutionsAny Matrix Size
Matrix Configuration
Set the matrix dimensions and enter your values
2
3
Matrix Input
Enter matrix elements (decimals and fractions allowed)
Example Matrices
Click on any example to load it into the calculator
2×3 Basic Example
Simple 2×3 matrix
[1, 2, 3]
[4, 5, 6]
3×3 Identity Leading
3×3 square matrix
[2, 4, 6]
[1, 3, 5]
[3, 7, 9]
Augmented Matrix (Linear System)
System of linear equations
[1, 2, 1, 8]
[3, 8, 1, 20]
[0, 4, 1, 8]
Inconsistent System
No solution example
[1, 1, 2]
[2, 2, 5]
What is Reduced Row Echelon Form?

A matrix is in Reduced Row Echelon Form (RREF) if it satisfies these conditions:

  • All nonzero rows are above any rows of all zeros
  • Each leading entry (pivot) of a row is 1
  • Each leading entry is to the right of the leading entry in the row above it
  • All entries in a column below and above a leading entry are zeros

Example RREF: Every matrix has a unique RREF form, making it perfect for solving linear systems.

Why Use RREF?

Key Applications:

  • Solve Linear Systems: Read solutions directly from RREF
  • Find Matrix Rank: Count non-zero rows in RREF
  • Determine Consistency: Check if system has solutions
  • Find Null Space: Identify free variables
  • Matrix Invertibility: Check if square matrix is invertible

Linear Systems: If the RREF of an augmented matrix has a row like [0 0 ... 0 | c] where c ≠ 0, the system is inconsistent.

Row Operations for RREF

Row Swapping

Ri↔RjR_i \leftrightarrow R_j

Exchange two rows to position pivots optimally.

Used to move the largest pivot element to the top for numerical stability.

Row Scaling

Ri=k⋅RiR_i = k \cdot R_i

Multiply a row by a non-zero constant k.

Used to make the leading entry (pivot) equal to 1.

Row Addition

Ri=Ri+k⋅RjR_i = R_i + k \cdot R_j

Add a multiple of one row to another row.

Used to eliminate entries above and below pivots.

Row Operations — Visual Examples
See how each elementary row operation transforms a matrix

1. Row Swap: R₁ ↔ R₂

Before
[1  2  3]
[4  5  6]
[7  8  9]
→
After
[4  5  6]
[1  2  3]
[7  8  9]

2. Row Scale: R₁ = (1/2)R₁

Before
[2  4  6]
[1  3  5]
→
After
[1  2  3]
[1  3  5]

3. Row Addition: R₂ = R₂ − R₁

Before
[1  2  3]
[1  3  5]
→
After
[1  2  3]
[0  1  2]
Learn More About RREF & Linear Algebra

MIT OpenCourseWare

Gilbert Strang's legendary Linear Algebra course (18.06)

Khan Academy

Video lessons on row echelon form and matrix elimination

LibreTexts

Free textbook on Gaussian elimination and row reduction

Frequently Asked Questions

What is RREF (Reduced Row Echelon Form)?
RREF is a canonical form where the leading entry in each row is 1, each leading 1 is the only nonzero entry in its column, and leading 1s move right as you go down. It is the 'simplest' form of a matrix.
What's the difference between REF and RREF?
REF (Row Echelon Form): leading entries don't need to be 1, and columns can have other nonzeros above pivots. RREF: leading entries are 1 and are the only nonzero entries in their column. RREF is unique; REF is not.
How do I find RREF?
Use Gaussian elimination with back-substitution: 1) Get a leading 1 in the first column, 2) Zero out entries below it, 3) Move to the next row and column, 4) Repeat, 5) Back-substitute to zero out entries above each pivot.
What can RREF tell me about a system?
RREF reveals the number of solutions (unique, infinite, or none), the rank of the matrix, a basis for the column and null spaces, and whether an n×n matrix is invertible (rank n). It is fundamental for solving Ax = b.
What are the elementary row operations?
Three operations that don't change the solution set: 1) Swap two rows, 2) Multiply a row by a nonzero scalar, 3) Add a multiple of one row to another. RREF is built using only these three operations.
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