What is reduced row echelon form?

A matrix is in reduced row echelon form when three things are true at once: the first non-zero entry of every non-zero row is a 1, that leading 1 is the only non-zero entry anywhere in its column, and each leading 1 sits further right than the one in the row above, with any all-zero rows pushed to the bottom. Every matrix has exactly one reduced row echelon form, whichever order you do the row operations in, which is what makes it worth comparing two people's answers against. This page reaches it by Gauss-Jordan elimination: pick a pivot, divide its row through to make the pivot 1, then subtract multiples of that row from every other row.

How is rref different from row echelon form?

Row echelon form only clears the entries below each pivot, so you are left with a triangle of numbers above the diagonal and you still have to back-substitute to read an answer. Reduced row echelon form clears above the pivots as well, so each pivot column ends up with a single 1 and nothing else, and the answer can be read straight off the last column. This tool always finishes the job: the operation is rref A, and it eliminates in every row rather than stopping halfway, so [1 2; 3 4] comes back as the identity rather than as [1 2; 0 1].

What do the pivots tell me?

The number of pivots is the matrix rank. In an augmented system, a non-pivot coefficient column corresponds to a free variable, while a pivot in the constants column can expose an inconsistency. Rank of A counts all entered columns, so it is the augmented rank when constants are included. For [1 2 3; 4 5 6; 7 8 9], RREF is [1 0 -1; 0 1 2; 0 0 0] and rank is 2. The linked System of Equations Calculator labels solution cases and free variables automatically.

Where does rounding show up in the answer?

Never in the arithmetic, and only in the display if you ask for it. Every entry is held as one whole number over another, so dividing a row by its pivot gives 1/3 and not 0.3333333333, and a row that should cancel to zero cancels to exactly zero instead of to 0.0000000000000002. Decimals you type are converted exactly on the way in as well: 0.5 becomes 1/2. The Display switch under the stage turns the same exact values into decimals for reading, at 0 to 10 places on the stepper, so 1/3 shows as 0.333 at three places while the value behind it is still a third.

Can I row reduce an augmented matrix to solve a system?

Yes, and that is the usual reason to want rref. Add one more column for the constants with the column stepper on grid A, so three equations in three unknowns become a 3 by 4 matrix. Reducing [2 1 -1 8; -3 -1 2 -11; -2 1 2 -3] gives [1 0 0 2; 0 1 0 3; 0 0 1 -1], which reads as x = 2, y = 3, z = -1. A rectangular matrix has no determinant, so that cell says the matrix is 3×4 and only a square matrix has one, while the rank beside it still counts the pivots.

Does it show each row operation?

No. What it shows is the finished reduced form, the rank of A, the determinant when the matrix is square, and a line saying the form was reached by exact row operations. The result copies out as plain text with aligned columns or as a LaTeX pmatrix, so it drops into a write-up where you show your own steps. If you want to follow the elimination yourself, the intermediate matrices are worth doing by hand and checking here: the answer this page gives is the unique reduced form, so any correct sequence of operations has to land on it.

Does opening this page change the main calculator?

The page selects its stated operation after saved restoration. Simply visiting it does not overwrite the saved draft. Only fields you deliberately edit update their stored values; changing a display preference does not save a preset matrix. Everything is computed in your browser.

RREF Calculator

Reduce a real numeric matrix up to 8×8 to reduced row echelon form using exact fractions. This page selects rref A after saved restoration. Rectangular and augmented matrices are supported: [2 1 -1 8; -3 -1 2 -11; -2 1 2 -3] gives [1 0 0 2; 0 1 0 3; 0 0 1 -1]. The rank is the rank of every entered column, including an augmented constants column. The page shows the finished reduced form, with text and LaTeX copies; it does not print intermediate operations or automatically parameterize a system.

A2 × 2

RowsColumns

1	0
0	1

Display

Reduced row echelon form: every pivot is 1 and is alone in its column.

Paste a whole matrix

Use one row per line or semicolon. Separate entries with spaces, tabs or commas; write fractions as 1/3. Brackets around each row are optional. Applying a matrix replaces that grid and its dimensions.

Matrix A text

Determinantnot zero, so A can be inverted
-2
Rank of Aall 2 rows independent; the most a 2×2 matrix can have
2
rref Aexact fractions
[ 1 0 ] [ 0 1 ]

Every pivot is 1 and is the only non-zero entry in its column, reached by exact row operations.

Every calculation uses exact rational numbers. Integers, decimals and fractions are read exactly. A singular square matrix has determinant 0 and no ordinary inverse. Fraction display keeps the exact value; Decimal rounds only the displayed result. Floating-point methods need care with rounding residuals near zero. This calculator accepts real numeric entries, not complex numbers or symbolic variables. It shows the finished answer rather than an intermediate row-operation trace.

Common questions

What is reduced row echelon form?: A matrix is in reduced row echelon form when three things are true at once: the first non-zero entry of every non-zero row is a 1, that leading 1 is the only non-zero entry anywhere in its column, and each leading 1 sits further right than the one in the row above, with any all-zero rows pushed to the bottom. Every matrix has exactly one reduced row echelon form, whichever order you do the row operations in, which is what makes it worth comparing two people's answers against. This page reaches it by Gauss-Jordan elimination: pick a pivot, divide its row through to make the pivot 1, then subtract multiples of that row from every other row.
How is rref different from row echelon form?: Row echelon form only clears the entries below each pivot, so you are left with a triangle of numbers above the diagonal and you still have to back-substitute to read an answer. Reduced row echelon form clears above the pivots as well, so each pivot column ends up with a single 1 and nothing else, and the answer can be read straight off the last column. This tool always finishes the job: the operation is rref A, and it eliminates in every row rather than stopping halfway, so [1 2; 3 4] comes back as the identity rather than as [1 2; 0 1].
What do the pivots tell me?: The number of pivots is the matrix rank. In an augmented system, a non-pivot coefficient column corresponds to a free variable, while a pivot in the constants column can expose an inconsistency. Rank of A counts all entered columns, so it is the augmented rank when constants are included. For [1 2 3; 4 5 6; 7 8 9], RREF is [1 0 -1; 0 1 2; 0 0 0] and rank is 2. The linked System of Equations Calculator labels solution cases and free variables automatically.
Where does rounding show up in the answer?: Never in the arithmetic, and only in the display if you ask for it. Every entry is held as one whole number over another, so dividing a row by its pivot gives 1/3 and not 0.3333333333, and a row that should cancel to zero cancels to exactly zero instead of to 0.0000000000000002. Decimals you type are converted exactly on the way in as well: 0.5 becomes 1/2. The Display switch under the stage turns the same exact values into decimals for reading, at 0 to 10 places on the stepper, so 1/3 shows as 0.333 at three places while the value behind it is still a third.
Can I row reduce an augmented matrix to solve a system?: Yes, and that is the usual reason to want rref. Add one more column for the constants with the column stepper on grid A, so three equations in three unknowns become a 3 by 4 matrix. Reducing [2 1 -1 8; -3 -1 2 -11; -2 1 2 -3] gives [1 0 0 2; 0 1 0 3; 0 0 1 -1], which reads as x = 2, y = 3, z = -1. A rectangular matrix has no determinant, so that cell says the matrix is 3×4 and only a square matrix has one, while the rank beside it still counts the pivots.
Does it show each row operation?: No. What it shows is the finished reduced form, the rank of A, the determinant when the matrix is square, and a line saying the form was reached by exact row operations. The result copies out as plain text with aligned columns or as a LaTeX pmatrix, so it drops into a write-up where you show your own steps. If you want to follow the elimination yourself, the intermediate matrices are worth doing by hand and checking here: the answer this page gives is the unique reduced form, so any correct sequence of operations has to land on it.
Does opening this page change the main calculator?: The page selects its stated operation after saved restoration. Simply visiting it does not overwrite the saved draft. Only fields you deliberately edit update their stored values; changing a display preference does not save a preset matrix. Everything is computed in your browser.

Exact: matrices are computed in rational arithmetic, so an inverse of an integer matrix is shown as fractions rather than rounded decimals, and a singular matrix is reported as singular instead of producing huge numbers.