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Reduced Echelon Form Calculator

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Reduced Row Echelon Form Calculator

Transform any matrix to its canonical RREF

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Matrix calculation illustration

Enter your matrix and click "Calculate RREF"

Online RREF calculator transforms matrices to canonical form with step-by-step solutions. Ideal for linear algebra, engineering, and scientific computing.

Reduced Row Echelon Form Calculator: Complete Guide

What is RREF?

The Reduced Row Echelon Form (RREF) is the simplified version of a matrix where:

  • Leading entries are 1 (pivots)
  • Pivots are right of those above
  • Zero rows are at bottom
  • Pivots are the only non-zero in columns

Key Formula

The Gauss-Jordan elimination algorithm:

1. Forward Elimination:
   a) Swap rows for non-zero pivot
   b) Scale pivot row to 1
   c) Eliminate below pivot
2. Backward Elimination:
   a) Eliminate above pivots

How to Use

  1. Input Matrix

    • Adjust dimensions (max 6×6)
    • Enter values or use presets

  2. Calculate

    • Automatic RREF computation
    • Visual transformation steps

  3. Interpret Results

    • Pivot positions highlighted
    • Solution space visualization

Technical Sources

  1. Algorithm Basis
    Gauss-Jordan elimination (19th century)
  2. Numerical Stability
    Partial pivoting implementation
  3. Precision Handling
    Floating-point with ε=1e-10 tolerance

Terminology

TermDefinition
PivotLeading 1 in row
RankNumber of pivots
Free VariableNon-pivot column variable
SingularMatrix with rank < dimension

FAQs

Q: Why does my matrix show decimal values?
A: Exact fractions aren’t always possible - we display rounded decimals for clarity.

Q: How to interpret no solution cases?
A: Inconsistent systems show [0 … 0 | 1] rows.

Q: Mobile device compatibility?
A: Fully responsive design works on all screens.

Important Notes

⚠️ Accuracy Considerations

  • Integer matrices may show decimals
  • Near-zero values (<1e-10) treated as zero

⚠️ Educational Use

  • Shows intermediate steps when enabled
  • Preserves original matrix for comparison

⚠️ Technical Limits

  • Maximum 6×6 matrices
  • Supports real numbers only
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