RREF Calculator
Transform matrices to Reduced Row Echelon Form (RREF) using the Gauss-Jordan elimination method. Enter your matrix below to see step-by-step row operations and analyze the matrix properties including rank, nullity, and solution space.
Input Matrix
RREF Result
Understanding RREF
Row Echelon Form
A matrix is in row echelon form when all nonzero rows are above zero rows and leading entries form a staircase pattern.
Reduced Form
RREF requires leading entries to be 1 and all other entries in pivot columns to be 0, creating a unique canonical form.
Row Operations
Three elementary operations: row swapping, row scaling, and row addition preserve matrix equivalence.
Matrix Rank
The number of nonzero rows in RREF equals the matrix rank, revealing the dimension of the column space.
Mathematical Theory & Applications
What is RREF?
Reduced Row Echelon Form (RREF) is the unique canonical form of a matrix obtained through Gauss-Jordan elimination. Every matrix has exactly one RREF, making it a powerful tool for analyzing linear systems, determining matrix rank, and finding basis vectors for vector spaces.
RREF Conditions:
1. All nonzero rows are above zero rows
2. Leading entry of each row is 1 (pivot)
3. Each pivot is to the right of pivots above
4. All entries above and below pivots are 0
Historical Development
The systematic approach to solving linear systems dates back to ancient Chinese mathematics, but the modern formulation was developed by Carl Friedrich Gauss and later refined by Wilhelm Jordan.
The Gauss-Jordan elimination method, which produces RREF directly, became fundamental to linear algebra and computational mathematics. It provides a systematic way to solve systems, find matrix inverses, and analyze linear transformations.
Row Operations
Row Swapping
Rᵢ ↔ Rⱼ
Exchange two rows to position pivots
Row Scaling
Rᵢ → cRᵢ (c ≠ 0)
Multiply row by nonzero constant
Row Addition
Rᵢ → Rᵢ + cRⱼ
Add multiple of one row to another
Pivot Creation
Make leading entry = 1
Scale row to create unit pivot
Column Clearing
Eliminate above and below pivot
Make all other column entries = 0
Zero Row Handling
Move zero rows to bottom
Maintain proper row ordering
Real-World Applications
System Solving
Determine if linear systems have unique, infinite, or no solutions
Matrix Inversion
Calculate matrix inverses using augmented matrix [A|I] → [I|A⁻¹]
Basis Finding
Identify linearly independent vectors and basis for vector spaces
Rank Determination
Find matrix rank for dimensionality analysis in data science
Computer Graphics
Solve transformation equations and projection matrices
Engineering Analysis
Circuit analysis, structural mechanics, and control systems
Frequently Asked Questions
Row Echelon Form (REF) only requires a staircase pattern with leading entries, while Reduced Row Echelon Form (RREF) additionally requires leading entries to be 1 and all other entries in pivot columns to be 0. RREF is the unique canonical form.
Rank is the number of linearly independent rows/columns, equal to the number of pivot positions in RREF. Nullity is the dimension of the null space, calculated as (number of columns - rank). Together, they satisfy the rank-nullity theorem.
Pivot columns contain the leading 1s in RREF and correspond to basic variables. Non-pivot columns correspond to free variables that can take any value. Free variables determine the solution space dimension for homogeneous systems.
RREF is unique because the conditions (leading 1s, zeros above and below pivots, staircase pattern) completely determine the form. Any sequence of valid row operations will produce the same RREF, making it a canonical representation.
RREF of the augmented matrix [A|b] immediately reveals the solution structure: inconsistent if a row [0...0|c] with c≠0 exists, unique solution if rank(A) = number of variables, or infinitely many solutions otherwise.
Yes! Form the augmented matrix [A|I] and reduce to RREF. If A is invertible, you'll get [I|A⁻¹]. If A is not invertible, you won't be able to achieve the identity matrix on the left side.
Zero rows in RREF appear at the bottom and indicate linear dependence among the original rows. The number of nonzero rows equals the matrix rank. Zero rows don't affect the solution but provide information about the matrix's properties.