Enter a matrix or an augmented system [A | b] and get its reduced row echelon form in exact fractions. Every swap, scaling and row replacement is shown with the matrix after it, along with pivot columns, rank, free variables and the solution or null space.
Last updated:
Matrix
×
Entries can be integers, decimals or fractions like -3/4. Empty cells count as 0.
Reduce to
Reduced row echelon form
–
1 pivot (leading 1)
Row operations, step by step
Advertisement
Advertisement
How to use the RREF Calculator
Set the size with the Rows and Cols menus, or press Paste matrix… to paste rows from a spreadsheet.
Type the entries: integers, decimals or fractions like −3/4. Empty cells count as 0.
Tick “Last column is the right-hand side” if you are solving a system of equations [A | b].
Press Calculate RREF to see the reduced matrix with highlighted pivots, the rank and the solution or null space.
Scroll through the steps to follow every row operation, or press Copy result to paste the answer elsewhere.
How Gauss–Jordan elimination reaches RREF
Start with the leftmost column that has a non-zero entry. If the top entry is 0, swap in a row below that has a non-zero entry.
Scale the pivot row so the pivot equals 1.
Replace every other row with itself minus a multiple of the pivot row so the rest of the column becomes 0. (Gaussian elimination for REF only clears entries below.)
Move one row down and one column right, and repeat until you run out of rows or columns.
Worked example
Solve x + 2y − z = −4, 2x + 3y − z = −11, −2x − 3z = 22. The augmented matrix and its RREF are:
Each variable column has a pivot, so the system has the unique solution x = −8, y = 1, z = −2. Load it with Examples → “3 equations, unique solution” to see all the steps.
Reading the result
What you see in the RREF
What it means
A pivot in every column of A (n pivots for n unknowns)
Unique solution; for a square matrix, A is invertible
A column of A without a pivot
That variable is free; infinitely many solutions (if consistent)
A row [0 … 0 | c] with c ≠ 0
Inconsistent system, no solution
Number of pivots
Rank of the matrix (rank + nullity = number of columns)
Identity matrix on the left of [A | I]
The right half is A⁻¹ (enter A and I side by side, untick right-hand side)
Frequently asked questions
What is reduced row echelon form (RREF)?
A matrix is in RREF when all zero rows are at the bottom, each non-zero row starts with a 1 (the pivot), every pivot is to the right of the pivot above it, and each pivot is the only non-zero entry in its column. Every matrix has exactly one RREF.
What is the difference between REF and RREF?
Row echelon form (REF) only needs zeros below each pivot, so a matrix can have many REFs. Reduced row echelon form also clears the entries above every pivot and scales pivots to 1, which makes it unique and lets you read solutions directly.
Which row operations are allowed?
Three elementary row operations: swap two rows (Ri ↔ Rj), multiply a row by a non-zero number (Ri → kRi), and add a multiple of one row to another (Ri → Ri + kRj). None of them changes the solution set of the system or the rank of the matrix.
How do I tell if a system has no solution or infinitely many?
Look at the RREF of the augmented matrix. A row like [0 0 … 0 | c] with c ≠ 0 means no solution. Otherwise, if every variable column has a pivot there is one unique solution; if some column has no pivot, that variable is free and there are infinitely many solutions.
How do I find the rank from the RREF?
The rank is the number of pivots, which equals the number of non-zero rows in the RREF. The pivot columns of the original matrix form a basis for its column space, and the number of non-pivot columns is the nullity.
Why does my textbook's answer look different?
The RREF itself is unique, so it should match exactly. Intermediate steps can differ because there are many valid orders of row operations; this calculator uses the standard Gauss–Jordan order (first non-zero entry as pivot, column by column).
Does the calculator round anything?
No. Every operation uses exact fractions with arbitrarily large integers, so 1/3 stays 1/3. Tick Show decimals if you prefer decimal approximations in the display.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.