RREF Calculator With Steps (Reduced Row Echelon Form)

Use this RREF calculator with steps to compute the reduced row echelon form of a matrix online. Get instant results with step-by-step row operations and detailed explanations.

RREF Calculator and Row Echelon Form Solver

This tool helps you find the reduced row echelon form (RREF) of a matrix using Gaussian elimination and Gauss-Jordan elimination.

RREF with steps: See each row operation clearly.
Row echelon form solver: Understand matrix simplification.
Online RREF calculator: Fast and free.

2. How to Compute RREF (Method)

The RREF is computed using the Gauss–Jordan elimination algorithm. Typical steps:

Step 1: Forward elimination (create zeros below pivots)

Use row operations to eliminate entries below each pivot.

Step 2: Normalize pivot to 1

$$ R_i := \frac{1}{a_{ii}} R_i $$

Step 3: Backward elimination (clear above pivots)

$$ R_j := R_j - k R_i $$ $$ k = a_{ji} $$

Repeat until every pivot is 1 and is the only nonzero element in its column.

3. Worked Examples

Example 1

Compute RREF for:

$$ A = \begin{bmatrix} 1 & 2 & -1 \\ 2 & 3 & 1 \\ -1 & -1 & 2 \end{bmatrix} $$

Perform row operations (summary):

$$ R_2 := R_2 - 2R_1 = [0, -1, 3] $$ $$ R_3 := R_3 + R_1 = [0, 1, 1] $$ $$ R_2 := -R_2 = [0, 1, -3] $$ $$ R_3 := R_3 - R_2 = [0, 0, 4] $$ $$ R_3 := \tfrac{1}{4}R_3 = [0,0,1] $$ $$ R_2 := R_2 + 3R_3 = [0,1,0] $$ $$ R_1 := R_1 + R_3 = [1,2,0] $$

Final RREF:

$$ \mathrm{RREF}(A) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Example 2

Compute RREF for:

$$ A = \begin{bmatrix} 1 & 3 & 3 \\ 2 & 6 & 9 \\ -1 & -3 & -3 \end{bmatrix} $$

Note: rows are proportional. Steps (summary):

$$ R_2 := R_2 - 2R_1 = [0,0,3] $$ $$ R_3 := R_3 + R_1 = [0,0,0] $$ $$ R_2 := \tfrac{1}{3}R_2 = [0,0,1] $$ $$ R_1 := R_1 - 3R_2 = [1,3,0] $$

Final RREF:

$$ \mathrm{RREF}(A) = \begin{bmatrix} 1 & 3 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} $$

4. Common Mistakes

Forgetting to make pivots equal to 1: RREF requires pivots = 1.
Only clearing below pivots: RREF requires clearing both above and below pivots.
Not swapping when pivot = 0: Swap with a lower row that has a nonzero entry.
Arithmetic/sign errors: Mistakes with fractions and signs break the process.
Incorrectly treating non-square matrices: RREF applies to rectangular matrices too — it is not limited to square matrices.

5. Practice Problems

Try the problems below. Click "Show Answer" to reveal the RREF results.

Exercise 1

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

Show Answer

$$ \mathrm{RREF} = \begin{bmatrix} 1 & 2 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$

Exercise 2

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

Show Answer

$$ \mathrm{RREF} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Exercise 3

$$ \begin{bmatrix} 1 & 4 & 2 \\ 2 & 5 & 3 \\ 4 & 9 & 6 \end{bmatrix} $$

Show Answer

$$ \mathrm{RREF} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Exercise 4

$$ \begin{bmatrix} 1 & 3 & 1 \\ 1 & 1 & -1 \\ 3 & 11 & 5 \end{bmatrix} $$

Show Answer

$$ \mathrm{RREF} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Frequently Asked Questions (FAQ)

Q: What is RREF in linear algebra?

RREF stands for reduced row echelon form, where each leading entry is 1 and all other entries in its column are zero.

Q: Does this RREF calculator show steps?

Yes. This calculator shows each row operation step by step so you can follow the full process.

RREF is closely connected with LU decomposition, eigenvectors, and matrix multiplication.

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Calculation Results

RREF Calculator With Steps (Reduced Row Echelon Form)

RREF Calculator and Row Echelon Form Solver

2. How to Compute RREF (Method)

Step 1: Forward elimination (create zeros below pivots)

Step 2: Normalize pivot to 1

Step 3: Backward elimination (clear above pivots)

3. Worked Examples

Example 1

Example 2

4. Common Mistakes

5. Practice Problems

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Frequently Asked Questions (FAQ)

Q: What is RREF in linear algebra?

Q: Does this RREF calculator show steps?

Navigation

Calculation Results

RREF Calculator With Steps (Reduced Row Echelon Form)

RREF Calculator and Row Echelon Form Solver

2. How to Compute RREF (Method)

Step 1: Forward elimination (create zeros below pivots)

Step 2: Normalize pivot to 1

Step 3: Backward elimination (clear above pivots)

3. Worked Examples

Example 1

Example 2

4. Common Mistakes

5. Practice Problems

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Frequently Asked Questions (FAQ)

Q: What is RREF in linear algebra?

Q: Does this RREF calculator show steps?

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