RREF Calculator with Steps
Type a matrix and watch it reduce one elementary row operation at a time, the way it is done on paper: pick the pivot, swap it into place, scale its row to a leading 1 and clear the rest of the column, with the pivot boxed in every matrix. Put a | bar before the last column and the augmented matrix is read back as a system of equations: a unique solution, a parametric family with free variables, or no solution at all. Choose row echelon form to finish with back substitution, the inverse to reduce [A | I] to [I | A⁻¹], or the determinant by elimination. Arithmetic is exact: fractions stay fractions.
Read the full guide to this tool
Results
Notes
- Three elementary row operations never change the solutions of a system: swap two rows, multiply a row by a non-zero number, and add a multiple of one row to another.
- In reduced row echelon form every pivot is 1, it is the only non-zero entry in its column, and each pivot sits to the right of the one above it. Every matrix has exactly one RREF, whichever pivots you choose on the way.
- A row reading 0 = c (c ≠ 0) means the system has no solution; a column without a pivot gives a free variable and infinitely many solutions.
- Determinants: a swap flips the sign, scaling a row by k scales the determinant by k, and adding a multiple of a row changes nothing.
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