RREF Calculator
Transform matrices to Reduced Row Echelon Form using Gaussian Elimination.
Final RREF Matrix
Rank & Nullity
Elimination Visualizer
Visual: Click Next to see how Gaussian Elimination turns the matrix into identity form step-by-step.
Understanding RREF
The Reduced Row Echelon Form (RREF) is the “simplest” possible version of a matrix. It is achieved by performing Elementary Row Operations until the matrix follows specific rules. It is the gold standard for solving systems of linear equations.
The 3 Golden Rules
- Leading 1s: The first non-zero number in any row must be a 1 (called a pivot).
- Staircase: Each leading 1 must be to the right of the one above it.
- Clean Columns: Any column with a leading 1 must have zeros everywhere else (above and below).
Row Operations
- Swap: Exchange two rows.
- Scale: Multiply a row by a non-zero number.
- Add: Add a multiple of one row to another row (to create zeros).
Why use RREF?
If you have a system of equations like:
2x + y = 5
x – y = 1
Converting its augmented matrix to RREF gives you the answer directly:
1 0 | 2 (means x = 2)
0 1 | 1 (means y = 1)
Frequently Asked Questions (FAQ)
Q: What is a Pivot?
A: A pivot is the leading 1 in a row. The number of pivots determines the Rank of the matrix (how many independent equations you truly have).
Q: Can RREF have fractions?
A: Yes! In fact, manual calculation often gets messy with fractions. This calculator preserves exact fractions (like 1/3) instead of decimals to be precise.
Q: What if a row is all zeros?
A: Rows of all zeros are pushed to the very bottom. This often happens if one equation is just a multiple of another (dependent).