🔄 Gauss-Jordan Elimination Calculator
Transform matrix to Reduced Row Echelon Form (RREF)
How to Use This Calculator
Select Matrix Size
Choose number of rows and columns for your matrix.
Enter Matrix Elements
Input all elements of your matrix (augmented matrix for systems of equations).
Calculate
Click to perform Gauss-Jordan elimination and get RREF.
Interpret Results
Read solutions from RREF: leading 1s indicate pivot positions, rank = number of pivots.
Formula
Reduced Row Echelon Form (RREF)
All leading entries are 1, leading 1s move right, rows of zeros at bottom
Gauss-Jordan Algorithm:
- Start with leftmost column, find pivot (largest absolute value)
- Swap rows to bring pivot to current row
- Normalize: divide pivot row by pivot value
- Eliminate: subtract multiples of pivot row from all other rows
- Move to next column, repeat until done
RREF Properties:
- Leading entry in each row is 1
- Leading 1 is the only non-zero entry in its column
- Leading 1s move right as you go down rows
- Rows of zeros are at the bottom
For Systems of Equations:
If last column represents constants, RREF gives solution: x₁ = value, x₂ = value, etc.
About Gauss-Jordan Elimination Calculator
The Gauss-Jordan Elimination Calculator transforms a matrix to its Reduced Row Echelon Form (RREF). This is an extension of Gaussian elimination that produces a matrix with leading 1s in each row and zeros above and below each leading 1. It's ideal for solving systems of linear equations.
When to Use This Calculator
- Solving Systems: Find solutions to Ax = b
- Finding Rank: Determine matrix rank
- Finding Inverse: Augment [A | I] to get [I | A⁻¹]
- Linear Independence: Check if vectors are linearly independent
- Basis Finding: Identify pivot columns for basis
Why Use Our Calculator?
- ✅ Complete RREF: Full reduced row echelon form
- ✅ Rank Display: Shows matrix rank
- ✅ Pivot Columns: Identifies linearly independent columns
- ✅ Accurate: Handles numerical precision
- ✅ Educational: Helps understand elimination process
- ✅ Free: No registration required
Key Concepts
- RREF: Unique form - every matrix has exactly one RREF
- Rank: Number of pivot columns = number of leading 1s
- Pivot: First non-zero entry in a row after elimination
- Leading 1: Pivot normalized to 1
- Free Variables: Columns without pivots in systems of equations
Applications
Solving Systems: Convert augmented matrix [A | b] to RREF to read solutions directly.
Matrix Inversion: Augment [A | I] and perform Gauss-Jordan to get [I | A⁻¹].
Frequently Asked Questions
What is RREF?
Reduced Row Echelon Form is a matrix form where: (1) leading entry in each row is 1, (2) leading 1 is only non-zero in its column, (3) leading 1s move right, (4) zero rows are at bottom. Every matrix has a unique RREF.
What's the difference between Gaussian and Gauss-Jordan elimination?
Gaussian elimination produces row echelon form (REF) with elimination only downward. Gauss-Jordan continues elimination upward, producing RREF with zeros above and below pivots.
How do I read solutions from RREF?
For augmented matrix [A | b], each row with a leading 1 gives an equation like x₁ = value. Columns without pivots represent free variables (can be any value).
Can RREF have different forms?
No! RREF is unique - every matrix has exactly one RREF, regardless of the elimination steps taken. This makes it useful for proofs and uniqueness.
What does rank tell me?
Rank = number of pivot columns = number of linearly independent rows = number of linearly independent columns = dimension of row space = dimension of column space.