Compute row echelon form of a matrix using Gaussian elimination. Paste matrix, press calculate, and see REF, RREF, rank, pivots.
;), columns by spaces or commas.
You can type expressions: 1/2, sqrt(2), -3^2. All rows must have equal length.
A ⟶ (row operations) ⟶ REF(A) ; then eliminate above pivots for RREF(A)
An Echelon Form Calculator converts a matrix to row echelon form (REF) or reduced row echelon form (RREF) using elementary row operations. In REF, each nonzero row begins with a leading entry (pivot) to the right of the pivot in the row above, and all entries below a pivot are zero. In RREF, pivots are \(1\) and are the only nonzero entries in their columns. With these forms you can read off the rank, detect consistency of linear systems, and obtain unique or parametric solutions directly from the augmented matrix. The calculator performs symbolic steps, keeping fractions exact and documenting every operation so learners can follow the logic.
The tool applies elementary operations—row swaps, row scaling by a nonzero constant, and row replacement \(R_i\leftarrow R_i+kR_j\)—to produce a staircase of pivots. From REF you may back-substitute to solve; from RREF you can read solutions immediately. The number of pivots equals \(\operatorname{rank}(A)\), which also equals the dimension of the column space. For an \(m\times n\) matrix, the nullity is \(n-\operatorname{rank}(A)\), describing free variables in a homogeneous system. For an augmented matrix \([A\mid \mathbf{b}]\), a pivot appearing in the last column with zeros in the corresponding row of \(A\) signals inconsistency.
Elementary row operations: \[ \begin{aligned} &\text{(Swap)}\ R_i \leftrightarrow R_j,\qquad \text{(Scale)}\ R_i \leftarrow c\,R_i\ (c\ne0),\\ &\text{(Replace)}\ R_i \leftarrow R_i + k\,R_j. \end{aligned} \]
Rank–nullity (for }A\in\mathbb{R}^{m\times n}\text{): \[ \operatorname{rank}(A) + \operatorname{nullity}(A) = n. \]
Consistency test (augmented matrix): \[ [A\mid \mathbf{b}] \ \text{in REF has a row } [0\ \cdots\ 0\mid c],\ c\ne0 \ \Rightarrow\ \text{inconsistent}. \]
System: \(\begin{cases} x+2y=5\\ 3x+8y=17\end{cases}\). Augmented matrix \(\left[\begin{smallmatrix}1&2&\vline&5\\3&8&\vline&17\end{smallmatrix}\right]\ \rightarrow\ \) RREF \(\left[\begin{smallmatrix}1&0&\vline&3\\0&1&\vline&1\end{smallmatrix}\right]\). Solution: \(x=3,\ y=1\). Rank \(=2\) (two pivots), no free variables.
\(\begin{cases} x+y=1\\ 2x+2y=3\end{cases}\). REF yields \(\left[\begin{smallmatrix}1&1&\vline&1\\0&0&\vline&1\end{smallmatrix}\right]\) \(\Rightarrow\) row \([0\ 0\mid1]\); inconsistent, no solution.
\(A=\begin{bmatrix}1&2&3\\2&4&6\\1&1&1\end{bmatrix}\ \rightarrow\ \) REF has two pivots \(\Rightarrow \operatorname{rank}(A)=2\). Since \(n=3\), \(\operatorname{nullity}(A)=1\). The homogeneous system \(A\mathbf{x}=\mathbf{0}\) has infinitely many solutions with one free parameter.
REF has zeros below pivots; RREF also scales pivots to 1 and clears above them, giving a canonical solution form.
The number of pivot positions equals \(\operatorname{rank}(A)\), the dimension of the column space.
If a row reduces to \([0\ \cdots\ 0\mid c]\) with \(c\ne0\), the system is inconsistent.
Non-pivot columns correspond to free variables; they parameterize infinitely many solutions of \(A\mathbf{x}=\mathbf{b}\) (when consistent).
No. Elementary row operations are equivalence transformations preserving the solution set of the linear system.