Row Echelon Form Calculator
Row Echelon Form Calculator transforms matrices using Gaussian elimination, identifying pivots, simplifying systems, and enabling rank, solution, and determinant insights.
[[1,2],[3,4]] or rows separated by new lines (values separated by commas or spaces). Expressions like 1/2, sqrt(2) are allowed. All rows must have the same number of columns.Results
A = [[1,2,3],[2,4,8],[1,1,2]]REF(A) = [[1,2,3],[0,1,2],[0,0,1]]Helping notes
- REF makes zeros below each leading 1. We scale each pivot row and eliminate entries below the pivot.
- Input may be a matrix like
[[1,2],[3,4]]or row-by-row values with commas/spaces. - Decimal places affect formatting only.
What is a Row Echelon Form Calculator?
A Row Echelon Form (REF) Calculator converts any matrix into an upper, step-like arrangement that exposes pivot positions and simplifies linear systems. For a matrix \(A\in\mathbb{R}^{m\times n}\), its REF satisfies: (i) all nonzero rows appear above zero rows; (ii) in each nonzero row, the first nonzero entry (the pivot) lies to the right of the pivot in the row above; (iii) all entries below each pivot are zero. Reduced row echelon form (RREF) adds: pivots equal \(1\) and each pivot is the only nonzero entry in its column. REF is central to solving \(A\mathbf{x}=\mathbf{b}\), computing rank, and understanding linear dependence.
About the Row Echelon Form Calculator
Given a matrix \(A=[a_{ij}]\), the calculator applies Gaussian elimination—systematic EROs—to zero out entries below each pivot and produce \(\operatorname{REF}(A)\). If desired, it can continue to RREF by back-substitution and column clearing. For a system \(A\mathbf{x}=\mathbf{b}\), the augmented matrix is $$[A\mid \mathbf{b}] = \begin{bmatrix} a_{11}&\cdots&a_{1n}&\vline& b_1\\ \vdots & &\vdots &\vline& \vdots\\ a_{m1}&\cdots&a_{mn}&\vline& b_m \end{bmatrix},$$ and the same EROs are applied to both \(A\) and \(\mathbf{b}\). The number of pivots equals the rank: \(\operatorname{rank}(A)=r\). Solutions follow from identifying pivot and free variables after REF/RREF.
How to Use this Row Echelon Form Calculator
- Enter matrix dimensions \(m\times n\), then fill entries of \(A\) (or the augmented matrix \([A\mid\mathbf{b}]\)).
- Choose the operation: REF or RREF. The tool applies EROs to create zeros below successive pivots.
- Read the output: pivots, rank \(r\), and (for augmented systems) whether the system is consistent or inconsistent.
- Interpret solutions: pivot columns give leading variables; non-pivot columns correspond to free variables and parametric solutions.
Consistency check uses the last row of \([A\mid\mathbf{b}]\): if a row becomes \([0\ \cdots\ 0\mid c]\) with \(c\neq 0\), the system is inconsistent.
Examples (applying the same formulas)
Example 1 (REF only): $$A=\begin{bmatrix} 1&2&1\\ 2&4&0\\ -1&-2&3 \end{bmatrix}.$$ Use pivot \(a_{11}=1\). Eliminate below: $$R_2\leftarrow R_2-2R_1=\begin{bmatrix}0&0&-2\end{bmatrix},\quad R_3\leftarrow R_3+R_1=\begin{bmatrix}0&0&4\end{bmatrix}.$$ REF: $$\operatorname{REF}(A)= \begin{bmatrix} 1&2&1\\ 0&0&-2\\ 0&0&4 \end{bmatrix},\quad \text{pivots in columns }1\text{ and }3,\ \operatorname{rank}(A)=2.$$
Example 2 (Augmented system to REF): $$\left[\begin{array}{cc|c} 2&1&5\\ 4&-2&2\\ -2&5&-1 \end{array}\right].$$ Pivot \(2\). Eliminate: $$R_2\leftarrow R_2-2R_1=\left[\begin{array}{cc|c}0&-4&-8\end{array}\right], \quad R_3\leftarrow R_3+R_1=\left[\begin{array}{cc|c}0&6&4\end{array}\right].$$ Next pivot \(-4\). Eliminate below: $$R_3\leftarrow R_3-\frac{6}{-4}R_2 =\left[\begin{array}{cc|c}0&0&16\end{array}\right].$$ REF: $$\left[\begin{array}{cc|c} 2&1&5\\ 0&-4&-8\\ 0&0&16 \end{array}\right],\quad \text{three pivots}\Rightarrow \operatorname{rank}(A)=2,\ \operatorname{rank}([A\mid\mathbf{b}])=3\ \Rightarrow\ \text{inconsistent.}$$