Dedicated matrix workspace

RREF Calculator — Reduced Row Echelon Form Solver

Solve a matrix in reduced row echelon form (RREF) and review every row-reduction step using exact fractions.

Rows
3
Columns
4

Up to 10 rows × 10 columns

Enter your matrix

Type one number in each box. Decimals and negative values are supported.

Formatted preview
[12−132419−1122]\begin{bmatrix}1 & 2 & -1 & 3 \\ 2 & 4 & 1 & 9 \\ -1 & 1 & 2 & 2\end{bmatrix}

Your row operations will appear here.

Set the matrix size, enter each value, then row reduce it to see every elementary operation.

Learn the method

What this calculator helps you notice

  1. 01Create a leading 1 in each pivot column
  2. 02Clear entries above and below each pivot
  3. 03Keep every row operation reversible

Row reduction guide

What is reduced row echelon form?

Reduced row echelon form, usually shortened to RREF, is a unique simplified form of a matrix. It makes pivot positions, free variables, inconsistent rows, and the solutions of a linear system easier to identify.

A matrix is in RREF when:

  • Every nonzero row starts with a leading 1.
  • Each leading 1 is the only nonzero entry in its column.
  • Leading 1s move to the right as you move down the rows.
  • Any all-zero rows appear at the bottom.

Worked example

Row reduce an augmented matrix

The augmented matrix below represents x + y = 5 and 2x − y = 1.

  1. Start[ 1 1 | 5 ] [ 2 −1 | 1 ]
  2. R₂ → R₂ − 2R₁[ 1 1 | 5 ] [ 0 −3 | −9 ]
  3. R₂ → −⅓R₂[ 1 1 | 5 ] [ 0 1 | 3 ]
  4. R₁ → R₁ − R₂[ 1 0 | 2 ] [ 0 1 | 3 ]x = 2 and y = 3

Common questions

Using the RREF solver

Can it handle fractions and decimals?

Yes. Enter integers, decimals, or fractions. The result uses exact fractions when possible so rounding does not hide a pivot or change the solution.

Can it row-reduce augmented matrices?

Yes. Include the constants as the last column. The resulting RREF shows whether a linear system has one solution, infinitely many solutions, or no solution.

What matrix sizes are supported?

The workspace supports rectangular and square matrices from 1 x 1 through 10 x 10. Adjust rows and columns independently before entering values.

Is RREF different from row echelon form?

Yes. Row echelon form clears entries below each pivot. RREF continues until every pivot is 1 and every other entry in each pivot column is zero.