# Row Echelon Form Calculator

Convert augmented matrices to row echelon form using Gaussian elimination. Shows step-by-step row operations and back substitution. Free REF calculator.

## What this calculates

Convert a 2x3 or 3x4 augmented matrix to row echelon form (REF) using Gaussian elimination. See every row operation, the final echelon form, and the solution obtained by back substitution.

## Inputs

- **System Size** — options: 2 equations, 2 unknowns (2x3 matrix), 3 equations, 3 unknowns (3x4 matrix) — Choose the size of the augmented matrix [A|b].
- **Row 1, Col 1**
- **Row 1, Col 2**
- **Row 1, Col 3** — For 2x3: this is the constant (b1). For 3x4: coefficient of x3.
- **Row 1, Col 4** — Only used for 3x4 matrices (constant b1).
- **Row 2, Col 1**
- **Row 2, Col 2**
- **Row 2, Col 3** — For 2x3: this is the constant (b2). For 3x4: coefficient of x3.
- **Row 2, Col 4** — Only used for 3x4 matrices (constant b2).
- **Row 3, Col 1** — Only used for 3x4 matrices.
- **Row 3, Col 2** — Only used for 3x4 matrices.
- **Row 3, Col 3** — Only used for 3x4 matrices.
- **Row 3, Col 4** — Only used for 3x4 matrices (constant b3).

## Outputs

- **Row Echelon Form** — formatted as text — The matrix in row echelon form.
- **Step-by-Step Operations** — formatted as text — The row operations performed to reach row echelon form.
- **Solution (via back substitution)** — formatted as text — The solution to the system, if one exists.
- **System Type** — formatted as text — Whether the system has a unique solution, infinitely many, or none.

## Details

Row echelon form (REF) is a simplified version of a matrix where every row starts with more leading zeros than the row above it. Gaussian elimination uses elementary row operations to reach this form.

**Row Echelon Form Properties:**

1. All zero rows are at the bottom
2. The leading entry (first non-zero number) of each non-zero row is to the right of the leading entry of the row above
3. All entries below a leading entry are zero

**Elementary Row Operations:**

- **Swap** two rows
- **Multiply** a row by a non-zero scalar
- **Add** a multiple of one row to another

These operations do not change the solution set of the system.

**Back Substitution:**

Once the matrix is in REF, solve from the bottom row up. The last row gives the value of the last variable directly. Substitute back to find the others.

**Example:**

For the system x + y = 5, 2x + 3y = 13:
Augmented matrix: [[1, 1, 5], [2, 3, 13]]
After R2 = R2 - 2*R1: [[1, 1, 5], [0, 1, 3]]
Solution: y = 3, x = 5 - 3 = 2

## Frequently Asked Questions

**Q: What is row echelon form?**

A: Row echelon form (REF) is a matrix where: (1) all zero rows are at the bottom, (2) the leading entry of each row is to the right of the one above, and (3) all entries below a leading entry are zero. It is the goal of Gaussian elimination and makes solving systems of equations possible via back substitution.

**Q: What is the difference between row echelon form and reduced row echelon form?**

A: Row echelon form (REF) requires zeros below each leading entry. Reduced row echelon form (RREF) additionally requires zeros above each leading entry and that each leading entry is 1. RREF gives the solution directly without back substitution, but REF is often sufficient and requires fewer operations.

**Q: What are elementary row operations?**

A: The three elementary row operations are: (1) swap two rows, (2) multiply a row by a non-zero constant, and (3) add a scalar multiple of one row to another row. These operations preserve the solution set of the system. Any matrix can be reduced to REF using a sequence of these operations.

**Q: What does it mean if a row becomes all zeros?**

A: If a row becomes all zeros (including the augmented column), it means one equation was a linear combination of the others (redundant). The system has infinitely many solutions (free variables). If a row is all zeros except the augmented entry (like [0, 0, 0, 5]), the system is inconsistent and has no solution.

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Source: https://vastcalc.com/calculators/math/row-echelon-form
Category: Math
Last updated: 2026-04-08