RREF Calculator
Reduce a matrix of up to 6 × 6 to reduced row echelon form by Gauss-Jordan elimination, with rank, pivot columns and row steps, as decimals or exact fractions. How it works
Use the RREF Calculator
About the RREF Calculator
A matrix of up to 6 rows and 6 columns reduces to reduced row echelon form (RREF) by Gauss-Jordan elimination, and the calculator shows the result with the rank and the pivot columns. The worked example 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3 reduces to the 3 × 3 identity with 2, 3 and −1 in the last column: x = 2, y = 3, z = −1, rank 3. Choose Exact fractions to see every entry in lowest terms instead of rounded decimals.
Matrices and what the calculator returns
| Matrix (rows separated by /) | Reduced form | Rank | Pivot columns | What it says |
|---|---|---|---|---|
| 2 1 -1 8 / -3 -1 2 -11 / -2 1 2 -3 | 1 0 0 2 / 0 1 0 3 / 0 0 1 -1 | 3 | 1, 2, 3 | x = 2, y = 3, z = -1 |
| 1 2 5 / 3 4 6 | 1 0 -4 / 0 1 4.5 | 2 | 1, 2 | x = -4, y = 4.5 |
| 1 1 1 4 / 1 -1 3 2 | 1 0 2 3 / 0 1 -1 1 | 2 | 1, 2 | Infinitely many solutions: x = 3 - 2z, y = 1 + z |
| 1 1 2 / 1 1 3 | 1 1 0 / 0 0 1 | 2 | 1, 3 | No solution: a pivot in the last column |
| 1 2 3 / 4 5 6 / 7 8 9 | 1 0 -1 / 0 1 2 / 0 0 0 | 2 | 1, 2 | The third row is 2 × row 2 - row 1 |
| 2 1 1 0 / 5 3 0 1 | 1 0 3 -1 / 0 1 -5 2 | 2 | 1, 2 | The right half is the inverse of the left half |
Each row of a matrix is typed into the grid; the first four rows read as augmented systems with the constants in the last column. Entries are shown as the calculator shows them.
How a typed entry is read
| Typed in a cell | Result | Reason |
|---|---|---|
| 1/2 | Read as 0.5 | A fraction |
| -3 / 4 | Read as -0.75 | A fraction; spaces around the slash are allowed |
| .25 | Read as 0.25 | A decimal |
| 1e-5 | Read as 0.00001 | Exponent form |
| 2.5E3 | Read as 2500 | Exponent form |
| 1,5 | Refused | Commas and letters are not read |
| 2x | Refused | Commas and letters are not read |
| 0x10 | Refused | Commas and letters are not read |
| 1/0 | Refused | The entry divides by zero |
| (empty) | Refused | An empty cell is not read as 0: enter 0 |
A refused entry is named by its row and column and no result is shown until it is fixed.
How to use this tool
- Set the number of rows and columns, 1 to 6 each. For a system of equations, use one column more than the number of unknowns and put the constants in the last column.
- Type each entry as a decimal, a fraction such as 1/2 or an exponent such as 1e-5. Type 0 for a zero.
- Read the reduced matrix with its rank and the pivot columns. They update as you type. Switch between Decimals and Exact fractions under the rank.
- Check the numbered row operations under it, and the Read as a system of equations panel, which treats the last column as the constants.
Frequently asked questions
- What is reduced row echelon form?
- A matrix is in RREF when each non-zero row starts with a 1 (the pivot), each pivot is the only non-zero entry in its column, each pivot is to the right of the pivot in the row above, and rows of zeros are at the bottom. 2 1 -1 8 / -3 -1 2 -11 / -2 1 2 -3 reduces to 1 0 0 2 / 0 1 0 3 / 0 0 1 -1.
- How do I solve a system of linear equations with RREF?
- Write the augmented matrix with the coefficients first and the constants in the last column, then reduce it. x + 2y = 5 and 3x + 4y = 6 become 1 2 5 / 3 4 6 and reduce to 1 0 -4 / 0 1 4.5, so x = -4 and y = 4.5. When the left part reduces to the identity, the last column is the solution.
- How do I find the rank of a matrix?
- The rank is the number of pivots in the reduced form, and the calculator prints it. 1 2 3 / 4 5 6 / 7 8 9 reduces to 1 0 -1 / 0 1 2 / 0 0 0, so its rank is 2 with pivots in columns 1 and 2: the third row is 2 × row 2 - row 1.
- How can I tell whether a system has no solution or many solutions?
- Look at the pivot columns. A pivot in the last column means no solution: 1 1 2 / 1 1 3 reduces to 1 1 0 / 0 0 1 with pivots in columns 1 and 3. A solvable system with fewer pivots than unknowns has infinitely many solutions: 1 1 1 4 / 1 -1 3 2 reduces to 1 0 2 3 / 0 1 -1 1, so x = 3 - 2z and y = 1 + z. The Read as a system of equations panel says this for you, naming the unknowns x1, x2 and x3: x1 = 3 − 2·x3, x2 = 1 + x3, with x3 free.
- Can I use it to find the inverse of a matrix?
- Yes, for a square matrix up to 3 × 3, since the matrix and the identity beside it must fit in 6 columns. Type 2 1 1 0 / 5 3 0 1 and it reduces to 1 0 3 -1 / 0 1 -5 2, so the inverse of 2 1 / 5 3 is 3 -1 / -5 2. If the left half does not become the identity, as with 1 2 1 0 / 2 4 0 1, the matrix has no inverse.
- Why does a small entry not show as 0?
- Entries show at most 4 decimals, but a value below 0.0001 keeps more digits so that it is not mistaken for zero: 1 0 0.00004 / 0 1 5 stays as it is. A value that is a whole number of thirds shows over 3: 1/3, 2/3, 4/3 or 5/3 (3 0 1 / 0 3 2 reduces to 1 0 1/3 / 0 1 2/3, and 3 4 reduces to 1 4/3). A value within 0.000000001 of a third, such as 0.3333333333, shows the same way. Other fractions show as decimals: 1/7 shows as 0.1429. Choose Exact fractions to see them in lowest terms: 3 1 1 / 1 2 1 (3x + y = 1, x + 2y = 1) reduces to 1 0 1/5 / 0 1 2/5 instead of 1 0 0.2 / 0 1 0.4.
Related Tools
View all toolsMatrix Calculator
Add, subtract and multiply matrices up to 10 × 10, or get the determinant, inverse, transpose, rank and row-reduced form. Answers are exact fractions.
System of Equations Solver
Solve 2 or 3 linear equations exactly, with Gauss-Jordan or Cramer’s rule steps and a graph of two lines. Says when there is no solution or infinitely many.
Eigenvalue Calculator
Eigenvalues and eigenvectors of a 2 × 2 or 3 × 3 matrix from its characteristic equation, with the trace and determinant and complex pairs a ± bi.
Determinant Calculator
Determinant of a square matrix up to 10 × 10 as an exact fraction and a decimal, with the cofactor working shown for 2 × 2 and 3 × 3.
Interpolation Calculator
Linear interpolation between two points: y at any x, x for a given y and the line y = mx + b, with a note when x lies outside them (extrapolation).
Slope Calculator
Enter two points for the slope (also as a fraction), y-intercept, distance, angle and line equation, or find a missing coordinate from a point and a slope.