What is reduced row echelon form?
A matrix is in reduced row echelon form (RREF) when:
- Any rows made entirely of zeros are at the bottom.
- The first nonzero entry in each nonzero row (its pivot) is
1. - Each pivot is to the right of the pivot in the row above.
- Every other entry in a pivot's column is
0.
Every matrix has exactly one RREF. This calculator finds it with Gauss–Jordan elimination, using three row operations: swapping two rows, multiplying a row by a nonzero number, and adding a multiple of one row to another.
How to use it
- Set the size, then type entries. Empty cells count as
0. - Use ↑ ↓ ← → and Enter to move between cells.
- Paste rows copied from a spreadsheet, or text separated by spaces or commas, into any cell to fill the grid.
- Turn on Augmented to treat the last column as the right-hand side of a system of equations. You'll see whether the system has one solution, infinitely many or none.
Why exact fractions?
Floating-point elimination builds up rounding errors, so an entry that should be 0 can come out as 0.0000000001. The rank and pivots it reports can then be wrong. This calculator uses rational arithmetic throughout, so the result is exact. Switch to Decimals only to change how the numbers are shown.
What the result tells you
- Rank is the number of pivots.
- Pivot columns of the original matrix form a basis for its column space.
- A square matrix is invertible exactly when its RREF is the identity matrix.