Transform any matrix to Row Echelon Form and Reduced Row Echelon Form with step-by-step row operations, pivot identification, rank determination, and visual pivot highlighting.
Row Echelon Form (REF) and Reduced Row Echelon Form (RREF) are canonical matrix representations obtained through elementary row operations. Every matrix can be transformed into these forms, and the process reveals the rank, pivot columns, free variables, and solution structure of associated linear systems.
A matrix is in REF when all zero rows are at the bottom, the leading entry (pivot) in each non-zero row is to the right of the pivot above, and all entries below each pivot are zero. RREF goes further: every pivot equals 1, and all entries above and below each pivot are zero, giving the unique reduced form.
This calculator accepts matrices up to 5×6 (for augmented systems) and performs the complete transformation to both REF and RREF. Each elementary row operation is recorded in a step-by-step log so you can follow the algorithm exactly as taught in textbooks. Pivot positions are color-highlighted in the result matrices, and a summary reports the rank, nullity, and pivot/free column classification.
The tool is valuable for students learning Gaussian elimination, instructors demonstrating the algorithm, and anyone verifying hand computations. Preset examples cover common textbook problems including systems with unique solutions, infinitely many solutions, and no solution.
Gaussian elimination involves dozens of row operations on a 3×4 or 4×5 matrix, and one arithmetic slip can propagate through every subsequent step. This calculator performs both REF and RREF with step-by-step displays of every swap, scale, and replacement operation. It identifies pivot columns, computes rank and nullity, and classifies the system (unique, infinite, or no solution). Six presets cover classic cases from textbooks. It is the most efficient way for students to verify homework and understand the algorithm.
Elementary row operations: Rᵢ ↔ Rⱼ (swap), kRᵢ → Rᵢ (scale), Rᵢ + kRⱼ → Rᵢ (replace). REF: zeros below pivots. RREF: pivots = 1, zeros above and below.
Result: RREF = [[1,0,0,2],[0,1,0,3],[0,0,1,-1]]
The 3×4 augmented matrix reduces to RREF revealing the unique solution x=2, y=3, z=−1. The rank is 3, nullity is 0, and all three columns are pivot columns.
Row Echelon Form (REF) requires all zero-rows at the bottom, each pivot strictly to the right of the one above, and zeros below every pivot. **Reduced** Row Echelon Form (RREF) additionally requires each pivot to be 1 and all entries above and below each pivot to be zero. REF is not unique — different row-operation sequences produce different echelon forms — but RREF is unique for any given matrix. Forward elimination produces REF; back-substitution (or continued elimination upward) produces RREF.
To solve Ax = b, form the **augmented matrix** [A|b] and row-reduce. If the last column is a pivot column (a row like [0 … 0 | c] with c ≠ 0), the system is **inconsistent** (no solution). If every column of A is a pivot column, the system has a **unique solution** readable directly from RREF. Otherwise, free variables exist, and the general solution is a **particular solution + null space**. This three-outcome classification is the core result of elementary linear algebra.
Gaussian elimination runs in O(n³) for an n×n system — the same cost as LU decomposition, which is essentially elimination stored in matrix form. In floating-point arithmetic, **partial pivoting** (always choosing the largest available pivot in each column) prevents small pivots from amplifying rounding errors. Scaled partial pivoting and complete pivoting further improve stability. For sparse or banded matrices, specialized elimination exploits structure for even faster solves. Despite iterative and direct alternatives, Gaussian elimination remains the most taught and most foundational algorithm in numerical linear algebra.
REF requires zeros below each pivot and pivots moving right. RREF additionally requires each pivot to be 1 and all entries above and below each pivot to be zero. RREF is unique for any given matrix.
REF reveals the rank of the matrix, identifies pivot and free columns, and allows back substitution to solve linear systems. RREF gives the solution directly without back substitution.
Three operations: swapping two rows, multiplying a row by a non-zero scalar, and adding a scalar multiple of one row to another. These operations do not change the row space or the solution set.
Yes — different sequences of row operations produce different REFs, but they all have the same pivot positions and rank. RREF, however, is unique.
A system is inconsistent if RREF contains a row like [0 0 0 ... | b] where b ≠ 0. This means the system has no solution.
The rank equals the number of non-zero rows in REF (equivalently, the number of pivots). Nullity = number of columns minus rank (for the coefficient matrix).