RREF Calculator (Row Echelon Form)

Reduce any matrix to reduced row echelon form with detailed step-by-step row operations. Enter your matrix and see every pivot, swap and elimination, then read off rank, nullity, pivot columns, free variables and solution type.

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About RREF Calculator (Row Echelon Form)

The RREF Calculator (Reduced Row Echelon Form Calculator) performs Gauss-Jordan elimination on any matrix, showing every row operation along the way. Whether you are solving a system of linear equations, finding the rank of a matrix, or identifying pivot and free columns, this tool gives you the complete step-by-step solution with exact fractional arithmetic — no rounding errors.

What Is Reduced Row Echelon Form (RREF)?

A matrix is in Reduced Row Echelon Form when it satisfies all of the following conditions:

1️⃣

Leading Ones

The first nonzero entry in each nonzero row is 1 (called a pivot).

⬇️

Staircase Pattern

Each pivot is to the right of the pivot in the row above it.

0️⃣

Zeros Above & Below

Every other entry in a pivot column is zero.

⬆️

Zero Rows at Bottom

Any rows consisting entirely of zeros are at the bottom.

How to Use the RREF Calculator

Step 1. Set the number of rows and columns using the +/− controls.

Step 2. Enter your matrix values into the grid cells. You can enter integers, decimals, or fractions like 1/3. Use Tab, Enter, or arrow keys to navigate between cells.

Step 3. If you are solving a system of equations, check Augmented [A|b] to mark the last column as the constants vector.

Step 4. Click Calculate RREF.

Step 5. Review the results: the RREF matrix, rank, nullity, pivot columns, and free variables. Use the step navigator or the Play button to watch each row operation unfold.

Row Echelon Form vs. Reduced Row Echelon Form

Property	REF (Row Echelon Form)	RREF (Reduced Row Echelon Form)
Zeros below pivots	Yes	Yes
Zeros above pivots	Not required	Yes
Pivots equal to 1	Not required	Yes
Uniqueness	Not unique	Unique for every matrix
Method	Forward elimination	Forward + back elimination

Understanding the Results

Rank is the number of pivot positions, representing the dimension of the column space (or row space). Nullity is the number of non-pivot columns, representing the dimension of the null space. The Rank-Nullity Theorem guarantees: Rank + Nullity = number of columns.

For augmented matrices \([A|b]\), the solution type depends on the RREF:

Condition	Solution Type
Row \([0\; 0\; \cdots\; 0\; \|\; c]\) with \(c \neq 0\)	No solution (inconsistent)
Every column of A is a pivot column	Unique solution
Some non-pivot columns in A, no inconsistent rows	Infinitely many solutions

Elementary Row Operations

The three operations used to compute RREF preserve the solution set of a linear system:

⇆

Row Swap

\(R_i \leftrightarrow R_j\) — exchange two rows.

Row Scaling

\(R_i \leftarrow kR_i\) — multiply a row by a nonzero scalar.

Row Addition

\(R_i \leftarrow R_i + kR_j\) — add a multiple of one row to another.

Frequently Asked Questions

What is Reduced Row Echelon Form (RREF)?

RREF is the canonical form of a matrix obtained through Gauss-Jordan elimination. In RREF, each leading entry (pivot) is 1, each pivot is the only nonzero entry in its column, and pivot positions move strictly to the right and down. Every matrix has a unique RREF.

What is the difference between REF and RREF?

Row Echelon Form (REF) only requires zeros below each pivot, while Reduced Row Echelon Form (RREF) additionally requires zeros above each pivot and all pivots equal to 1. RREF is unique for any given matrix, but REF is not.

How do you find the rank of a matrix using RREF?

The rank of a matrix equals the number of pivots (leading 1s) in its RREF. Pivot columns are the columns containing these leading 1s. The nullity equals the number of columns minus the rank, which is also the number of free variables.

How do you solve a system of equations using RREF?

Write the augmented matrix [A|b] for the system Ax = b, then reduce it to RREF. If any row has the form [0 0 ... 0 | c] with c nonzero, the system is inconsistent (no solution). Otherwise, pivot columns give determined variables and non-pivot columns correspond to free variables that can take any value.

What row operations are used to find RREF?

Three elementary row operations are used: (1) swapping two rows, (2) multiplying a row by a nonzero scalar, and (3) adding a multiple of one row to another. These operations do not change the row space or solution set of a linear system.

Reference this content, page, or tool as:

"RREF Calculator (Row Echelon Form)" at https://MiniWebtool.com/rref-calculator-row-echelon-form/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-09

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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