Row Echelon Form Calculator
Matrix Input
How to Use
- Set matrix dimensions (up to 6x6)
- Enter matrix values or click "Randomize"
- Click "Calculate REF" to transform to row echelon form
- Toggle options to show steps or visualization
What is Row Echelon Form?
A matrix is in REF when:
- All zero rows are at the bottom
- The leading coefficient (pivot) of each non-zero row is 1
- Each pivot is to the right of the pivot in the row above
Row Echelon Form Calculator instantly transforms matrices using Gaussian elimination. Perfect for students and engineers. Step-by-step solutions + visual charts.
Row Echelon Form Calculator: Instant Matrix Solutions
What Is It?
A Row Echelon Form (REF) Calculator transforms any matrix into its simplified row echelon form using Gaussian elimination. Essential for linear algebra, equation solving, and matrix analysis.
Key Formula
Gaussian Elimination Steps:
- Pivot Selection: Identify leftmost non-zero column
- Row Swapping: Move rows to position pivots
- Row Operations: Create zeros below pivots using:
Row₂ = Row₂ - (Factor × Row₁)
How to Use
- Enter your matrix values (up to 6×6)
- Click “Calculate REF”
- View results:
- Simplified matrix
- Step-by-step operations (optional)
- Visual chart (optional)
FAQs
Q: Why can’t I get a perfect triangular form?
A: Some matrices (like singular ones) won’t form perfect triangles due to zero rows.
Q: What’s the difference between REF and RREF?
A: REF has leading 1s with zeros below, while RREF (Reduced REF) adds zeros above pivots too.
Key Terms
- Pivot: First non-zero element in a row
- Leading Coefficient: Leftmost non-zero number in a row
- Gaussian Elimination: The algorithm used to achieve REF
Algorithm Source
Based on standard linear algebra procedures from Gilbert Strang’s “Introduction to Linear Algebra” (MIT curriculum).
Important Notes
- Only works with numerical values (no variables)
- Rounding may occur with fractions (e.g., 0.333 → 1/3)
- For RREF, use our “Advanced Matrix Calculator”