MathIsimple
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.

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