What is the difference between REF and RREF?

In row echelon form (REF) each leading entry (pivot) is to the right of the one above and everything below a pivot is zero — the result of Gaussian elimination. Reduced row echelon form (RREF) additionally makes every pivot 1 and clears the rest of each pivot column — the result of Gauss-Jordan elimination. A matrix has exactly one RREF but many possible REFs.

How do row operations change the determinant?

Swapping two rows flips its sign, adding a multiple of one row to another leaves it unchanged, and multiplying a row by k multiplies it by k. The calculator uses only swaps and row additions to reach an upper-triangular matrix, so det(A) = (product of the diagonal) × (−1)^(number of swaps), and it checks the answer with cofactor expansion.

How do you find an inverse with row operations?

Write A next to the identity matrix as [A | I] and row-reduce the whole thing until the left half becomes I; the right half is then A⁻¹. If a row of zeros appears on the left, det(A) = 0 and A has no inverse.

How can I tell whether a linear system has a solution?

Reduce the augmented matrix [A | b] and compare ranks. If rank(A) < rank([A | b]) a row reads 0 = non-zero and there is no solution. If the ranks are equal and match the number of unknowns the solution is unique; if they are equal but smaller, there are infinitely many solutions and the variables without pivots are free parameters.

🪜

🪜RREF Calculator with Steps - Row Reduction, Det & Inverse

When you learn linear algebra the working matters more than the answer. Enter a matrix up to 6 × 6 and the calculator row-reduces it one elementary row operation at a time, writing each step in textbook form — R2 ← R2 − 3R1, R1 ↔ R2, R1 ← R1 ÷ 2 — followed by the matrix after that step, with the pivot and the changed rows highlighted. It reduces to row echelon form (Gaussian elimination) or reduced row echelon form (Gauss-Jordan), computes the determinant both by elimination and by cofactor (Laplace) expansion, finds the inverse by reducing [A | I], gives the rank and pivot columns, and solves Ax = b from the augmented matrix, reporting a unique solution, no solution or the general solution with free variables. Everything uses exact fractions, so 1/3 never turns into 0.3333.

Free · No sign-up · Runs in your browserLast updated: 2026-10-26

Loading tool...

How to use Matrix Step-by-Step

1Choose the operation: RREF, row echelon form, determinant, inverse, rank or solve Ax = b
2Set the size and fill in the matrix (integers, fractions like 3/4 or decimals; you can paste a block from a spreadsheet)
3Follow each row operation and the resulting matrix down to the final answer

FAQ

What is the difference between REF and RREF?: In row echelon form (REF) each leading entry (pivot) is to the right of the one above and everything below a pivot is zero — the result of Gaussian elimination. Reduced row echelon form (RREF) additionally makes every pivot 1 and clears the rest of each pivot column — the result of Gauss-Jordan elimination. A matrix has exactly one RREF but many possible REFs.
How do row operations change the determinant?: Swapping two rows flips its sign, adding a multiple of one row to another leaves it unchanged, and multiplying a row by k multiplies it by k. The calculator uses only swaps and row additions to reach an upper-triangular matrix, so det(A) = (product of the diagonal) × (−1)^(number of swaps), and it checks the answer with cofactor expansion.
How do you find an inverse with row operations?: Write A next to the identity matrix as [A | I] and row-reduce the whole thing until the left half becomes I; the right half is then A⁻¹. If a row of zeros appears on the left, det(A) = 0 and A has no inverse.
How can I tell whether a linear system has a solution?: Reduce the augmented matrix [A | b] and compare ranks. If rank(A) < rank([A | b]) a row reads 0 = non-zero and there is no solution. If the ranks are equal and match the number of unknowns the solution is unique; if they are equal but smaller, there are infinitely many solutions and the variables without pivots are free parameters.