How row reduction works
Only three operations are allowed, and none of them changes the solutions of the system:
- Swap two rows.
- Scale a row by a nonzero number.
- Add a multiple of one row to another.
Gauss–Jordan elimination works column by column: find a pivot, scale it to 1, then clear every other entry in that column.
Example system
x + 2y − z = −4
2x + 3y − z = −11
−2x − 3z = 22
As an augmented matrix this is the default example above. Its RREF reads x = −8, y = 1, z = −2.
Reading the result
- One solution: a pivot in every variable column.
- No solution: a row of zeros with a nonzero answer, such as [0 0 0 | 3].
- Infinitely many: at least one variable column without a pivot. Those variables are free.
Need the inverse or determinant instead? Use the matrix calculator.
Last reviewed October 2026.
Frequently asked questions
What is reduced row echelon form (RREF)?
A matrix is in RREF when every leading entry (pivot) is 1, each pivot is to the right of the one above it, every pivot is the only nonzero number in its column, and any rows of all zeros are at the bottom.
How does RREF solve a system of equations?
Write the system as an augmented matrix (coefficients plus the constants in the last column) and reduce it. Each pivot row then reads directly as variable = value. Tick "Last column is the answer column" and the calculator reads the solution for you.
Why does the calculator show fractions?
It uses exact fraction arithmetic, so 1/3 stays 1/3 instead of becoming 0.333333. That avoids rounding errors and matches what you would get by hand.
What if the system has no solution or infinitely many?
A row like [0 0 0 | 5] means 0 = 5, so there is no solution. If there are fewer pivots than variables (and no contradiction), some variables are free and there are infinitely many solutions. The calculator tells you which case applies.
What is the rank of a matrix?
The rank is the number of pivots in the RREF, which equals the number of independent rows. A square matrix has full rank exactly when it is invertible.