RREF Calculator
Enter a matrix, one row per line, to reduce it to reduced row echelon form with every row operation shown.
Separate numbers with spaces or commas. For a system, include the constants as the last column.
Reduced row echelon form (RREF) is the simplest version of a matrix reached by Gauss-Jordan elimination. Each leading entry is 1, sits to the right of the one above it, and is the only non-zero number in its column. This calculator row-reduces any matrix and shows each row operation.
How to Use This Calculator
Type your matrix with one row on each line, separating the numbers in a row by spaces or commas. Every row must have the same number of entries. To solve a system of equations, write the augmented matrix — the coefficients followed by the constant term as the final column of each row.
The calculator returns the reduced row echelon form and lists every row operation it performed, so you can follow the Gauss-Jordan process rather than just see the answer.
How the Calculation Works
Gauss-Jordan elimination works column by column. It finds a pivot — a leading non-zero entry — swaps it into position if needed, divides its row so the pivot becomes 1, then subtracts the right multiple of that row from every other row to make the rest of the column zero.
Repeating this for each pivot column leaves a matrix where every leading 1 is alone in its column and steps to the right as you go down. That is the reduced row echelon form, and it is unique for any given matrix.
for each pivot: scale it to 1, then subtract multiples to clear its column
Worked Example
Take the matrix with rows 1 2 3 and 4 5 6. Subtract 4 times row 1 from row 2 to get 0 −3 −6. Divide row 2 by −3 to get 0 1 2. Finally subtract 2 times the new row 2 from row 1 to get 1 0 −1.
The RREF is the rows 1 0 −1 and 0 1 2. If this were an augmented system, it would read x = −1 and y = 2.
Assumptions and Limitations
- Every row must have the same number of columns
- Entries are treated as decimals; results round very small values to zero
- For a system, include the constants as the final column (augmented matrix)
- The RREF of a matrix is unique regardless of the steps taken to reach it
When to Use an RREF Calculator
Linear-algebra students use this to row-reduce a matrix to reduced row echelon form step by step. Run it to solve a system of equations, find a matrix's rank, or check by-hand row operations. Seeing each elimination step makes it a strong study tool as well as a quick answer check.
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Open calculator →Frequently Asked Questions
What is reduced row echelon form?
It is the simplest form of a matrix from Gauss-Jordan elimination: every leading entry is 1, each leading 1 is the only non-zero number in its column, and the leading 1s move right as you go down the rows.
How do you find the RREF of a matrix by hand?
Use Gauss-Jordan elimination: for each column, get a leading 1 (swapping and scaling rows as needed), then subtract multiples of that row from all others to zero out the rest of the column. Repeat for each pivot.
What is the difference between REF and RREF?
Row echelon form (REF) only needs zeros below each leading entry. Reduced row echelon form (RREF) also requires the leading entries to be 1 and zeros above them too, making it fully reduced and unique.
How do I solve a system with RREF?
Write the augmented matrix with the constants as the last column, reduce it to RREF, and read the solution directly: each leading 1 gives the value of its variable in the final column.
Is the RREF of a matrix unique?
Yes. While the steps to get there can vary, every matrix has exactly one reduced row echelon form, which is why it is used as a canonical form in linear algebra.
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