RREF Calculator – Get Reduced Row Echelon Form of Any Matrix

Enter a matrix to instantly get its Reduced Row Echelon Form (RREF) along with a step-by-step solution of the row operations.

What is the RREF Calculator?

RREF Calculator, also known as RREF Solver, is used to reduce a matrix into its Reduced Row Echelon form by performing row operations (Gauss–Jordan elimination). It performs elementary row operations to create pivot 1s and zeros above and below each pivot.

It also shows how the row operations were performed at each step so that you can understand the concept and the process clearly.

How To Use the RREF Matrix Calculator?

Follow the following steps to use the RREF Matrix Calculator effectively.

  1. Enter the Matrix
    – Input the size of your matrix (e.g., 2×2, 3×3, 4×4, etc.).
    – Fill in the values of the matrix in the given input boxes.
  2. Click the Calculate Button
    – The calculator will display the matrix in its Reduced Row Echelon Form (RREF).
    – You’ll also see the step-by-step row operations, so you can follow exactly how the matrix was reduced.
  3. Interpret the Answer
    If you’re solving a system of linear equations, the RREF form makes it easy to identify:
    –  A unique solution
    – Infinite solution
    – or No solution

What is the Reduced Row Echelon Form of the Matrix?

The reduced row echelon form of a matrix is a special kind of matrix you get by performing Gaussian elimination (row reduction) until the matrix is in its simplest form.

It satisfies the conditions:

Leading 1s (pivots):

In each nonzero row, the first nonzero entry (from left to right) is 1. This is called a leading 1.

Zeros above and below pivots:

Each leading 1 is the only nonzero entry in its column. That means everything above and below it is zero.

Stair-step pattern:

Each leading 1 in a lower row appears to the right of the leading 1 in the row above it (like steps going down to the right).

Zero rows at the bottom:

Any rows made entirely of zeros are at the bottom of the matrix.

Example:

Suppose we start with this matrix:

\begin{bmatrix}
1 & 2 & -1 \\
2 & 4 & 3 \\
3 & 6 & 0
\end{bmatrix}

After row reduction, RREF is:

\begin{bmatrix}
1 & 2 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{bmatrix}

How to Transform a Matrix to Reduced Row Echelon Form?

The following are the 5 easy and detailed steps to transform the matrix to reduced row echelon form.

1. Choose Pivot → Find the leftmost nonzero column, make the top element 1 (swap/scale if needed).

2. Zero Below Pivot → Use row operations to make all entries below the pivot 0.

3. Move to Next Pivot → Repeat for the next pivot to the right.

4. Zero Above Pivots → Make all entries above each pivot 0.

5. Check RREF → Pivots are 1, pivot columns have zeros elsewhere, and zero rows are at the bottom.

Example:

Let’s apply these 5 steps to the following matrix:

\begin{bmatrix}
1 & 2 & -1 \\
2 & 4 & 3 \\
3 & 6 & 0
\end{bmatrix}

Step 1: Choose Pivot

The leftmost nonzero column is column 1. The top element is already 1, so no swapping or scaling is needed.

Pivot = 1 (row 1, column 1) ✓

Step 2: Zero Below Pivot

Eliminate the entries below the pivot in column 1:

  • R2 → R2 − 2×R1: [2−2(1), 4−2(2), 3−2(−1)] = [0, 0, 5]
  • R3 → R3 − 3×R1: [3−3(1), 6−3(2), 0−3(−1)] = [0, 0, 3]

Matrix now looks like:

\begin{bmatrix}
1 & 2 & -1 \\
0 & 0 & 5 \\
0 & 0 & 3
\end{bmatrix}

Step 3: Move to Next Pivot

Column 2 is all zeros below row 1, so we skip it. Move to column 3. The top available entry is 5 (row 2). Scale row 2 to make it 1:

  • R2 → R2 ÷ 5: [0, 0, 1]

Matrix now looks like:

\begin{bmatrix}
1 & 2 & -1 \\
0 & 0 & 1 \\
0 & 0 & 3
\end{bmatrix}

Now eliminate below this pivot:

  • R3 → R3 − 3×R2: [0, 0, 3−3(1)] = [0, 0, 0]

Matrix now looks like:

\begin{bmatrix}
1 & 2 & -1 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{bmatrix}

Step 4: Zero Above Pivots

Eliminate the entry above the pivot in column 3 (row 1, column 3 = −1):

  • R1 → R1 + 1×R2: [1, 2, −1+1(1)] = [1, 2, 0]

Matrix now looks like:

\begin{bmatrix}
1 & 2 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{bmatrix}

Step 5: Check RREF

✓ Each pivot is 1
✓ Each pivot column has zeros above and below it
✓ Zero row is at the bottom
✓ Pivots follow a stair-step pattern left to right

This is the final RREF:

\begin{bmatrix}
1 & 2 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{bmatrix}

What is the Difference Between Row Echelon & Reduced Echelon Form?

Row Echelon Form (REF) is a partly simplified stair-step form, while Reduced Row Echelon Form (RREF) goes further by making each pivot 1 and the only nonzero in its column, giving a fully simplified and unique matrix.

Matrix is in Row Echelon Form (REF) if :

  1. All nonzero rows are above any rows of all zeros.
  2. Each pivot (first nonzero entry in a row) is to the right of the pivot in the row above it (stair-step pattern).
  3. All entries below each pivot are zero.

Matrix is in Reduced Row Echelon Form (RREF) if:

  1. It satisfies all REF rules.
  2. Each pivot is 1.
  3. Each pivot is the only nonzero entry in its column (zeros above and below it).

Frequently Asked Questions (FAQs)

Yes, every matrix has an RREF (Row Reduced Echelon Form). The Gauss–Jordan elimination algorithm guarantees that if you keep applying row operations, you’ll eventually reach a unique RREF

There are 4 rules for RREF


1. Leading 1s (pivots): If a row has nonzero entries, the first nonzero entry in that row must be 1.
2. Zeros above and below pivots: Each pivot must be the only nonzero entry in the column.
3. Pivot position order: As you move down the rows, each pivot must appear to the right of the pivot in the row above it.
4. Zero rows at the bottom: If a row is entirely zero, it must be placed below all the nonzero rows

The following are the uses of the RREF Calculator

  • System of Linear Equations
  • Inverse of a Matrix
  • Rank of a Matrix
  • Linear Independence

A zero row in RREF is just a row of all zeros, always placed at the bottom of the matrix. It shows that one equation (or more) didn’t add new information.