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RREF Calculator

Reduced row echelon form is where Gauss-Jordan elimination stops: every pivot is 1, every pivot is the only non-zero entry in its column, and any row that was a combination of the others has become a row of zeros. This page opens the matrix calculator with rref A already selected, so you type the matrix and the reduced form appears as you type, up to 8 rows by 8 columns and rectangular when you are reducing an augmented system. The row operations run in exact fractions rather than decimals, so [1 2 -1 3; 2 4 1 9; 3 6 2 14] reduces to [1 2 0 4; 0 0 1 1; 0 0 0 0] with no stray 0.9999 anywhere in it, and the number of pivots is reported beside the answer as the rank of A.

A2 × 2
RowsColumns
10
01
Display

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

  • Determinantnot zero, so A can be inverted
    -2
  • Rank of Athe 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 entry is a fraction, not a decimal. Numbers here are held as one whole number over another and are added, multiplied and divided without rounding at any step, so the inverse of an integer matrix arrives as fractions and stays readable: 1/3 is 1/3, not 0.3333333333. It also fixes the failure that matters most. A singular matrix has determinant exactly 0, but in decimals it usually does not: the determinant of [1 2 3; 4 5 6; 7 8 9] comes out at about 0.00000000000000067, so a calculator that trusts that number inverts the matrix anyway and hands back entries around a thousand trillion. This one gets 0, and says the matrix is singular. Switch the display to Decimal whenever a decimal is what you need: it rounds the same exact values for reading, and never for calculating.

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?
Count them and you have the rank, which the page prints as Rank of A beside the result along with how many rows could be built out of the others. A pivot column is a column that is independent of the ones before it; a column with no pivot corresponds to a free variable in a system of equations. For a square matrix the two ends meet: reducing to the full identity is exactly the case where the rank equals the size and the determinant is not zero, and the page says so in words. Reduce [1 2 3; 4 5 6; 7 8 9] and you get [1 0 -1; 0 1 2; 0 0 0], rank 2, with the determinant reported as zero and the matrix named as singular.
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?
It sets the operation to rref A for this page ahead of whatever the calculator last remembered, and it leaves that memory alone: go back to the full matrix calculator and it is still on the operation you left it on. The matrix you type does travel with you, since it is usually the one you are still working on.

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.