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RREF Calculator (Row Echelon Form)

Reduce any matrix to reduced row echelon form (RREF) with detailed step-by-step row operations. Enter your matrix, see every pivot, swap, and elimination step visualized. Identify rank, nullity, pivot columns, free variables, and solution type instantly.

RREF Calculator (Row Echelon Form)
Examples:
Rows: 3
×
Cols: 3
(
)

Embed RREF Calculator (Row Echelon Form) Widget

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

PropertyREF (Row Echelon Form)RREF (Reduced Row Echelon Form)
Zeros below pivotsYesYes
Zeros above pivotsNot requiredYes
Pivots equal to 1Not requiredYes
UniquenessNot uniqueUnique for every matrix
MethodForward eliminationForward + 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:

ConditionSolution Type
Row \([0\; 0\; \cdots\; 0\; |\; c]\) with \(c \neq 0\)No solution (inconsistent)
Every column of A is a pivot columnUnique solution
Some non-pivot columns in A, no inconsistent rowsInfinitely 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

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

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