Reduced Row Echelon Form Calculator
Solve systems of linear equations using Gauss-Jordan elimination with step-by-step solutions
System of Linear Equations
Augmented Matrix
Enter the coefficients and constants for your system of equations. The rightmost column represents the constants.
Reduced Row Echelon Form (RREF)
Solution Analysis
The system has a unique solution
Step-by-Step Solution
Step 1: Initial augmented matrix
Step 2: R2 → R2 + (1.000) × R1
Step 3: R2 → (1/3.000) × R2
Step 4: R3 → R3 + (1.000) × R2
Step 5: R3 → (1/2.333) × R3
Step 6: R1 → R1 + (-1.000) × R3
Step 7: R2 → R2 + (-0.333) × R3
Step 8: R1 → R1 + (-1.000) × R2
Example System
Sample System of Equations
Consider the system:
-x + 2y = 25
-y + 2z = 16
Solution Process
1. Convert to augmented matrix form
2. Apply elementary row operations
3. Obtain row echelon or reduced row echelon form
4. Read the solution from the final matrix
Elementary Row Operations
Row Swapping
R₁ ↔ R₂
Exchange any two rows
Row Scaling
R₁ → kR₁
Multiply row by non-zero constant
Row Addition
R₁ → R₁ + kR₂
Add multiple of one row to another
Applications
Solving systems of linear equations
Finding matrix rank and nullity
Determining linear independence
Computing matrix inverse
Finding basis for vector spaces
Understanding Row Echelon Forms
What is Row Echelon Form?
Row Echelon Form (REF) is a simplified form of a matrix obtained through elementary row operations. A matrix is in REF if all non-zero rows are above rows of all zeros, and each leading entry is to the right of the leading entry in the row above.
REF Properties
- •All non-zero rows are above zero rows
- •Leading entries form a "staircase" pattern
- •All entries below leading entries are zero
Reduced Row Echelon Form (RREF)
RREF is a further simplified form where each leading entry is 1, and all other entries in its column are 0. This form makes it easier to read solutions directly from the matrix.
RREF Additional Properties
- •All leading entries are 1 (called pivots)
- •All other entries in pivot columns are 0
- •Solutions can be read directly from the matrix
Tip: RREF is unique for any matrix, while REF is not unique.