What is the difference between Gaussian elimination and Gauss-Jordan elimination?

Gaussian elimination reduces a matrix to row echelon form (upper triangular) and then uses back-substitution to find the solution. Gauss-Jordan elimination goes further — it reduces the matrix all the way to Reduced Row Echelon Form (RREF) by eliminating entries both above and below each pivot. This means you can read the solution directly without back-substitution.

How do I do Gauss-Jordan elimination step by step?

Write the augmented matrix [A|b]. Find the leftmost non-zero column and use partial pivoting to place the largest value as the pivot. Divide the pivot row to make the pivot equal to 1. Subtract multiples of the pivot row from all other rows to zero out the rest of the pivot column. Move to the next column and repeat until you reach RREF.

Can Gauss-Jordan elimination find the inverse of a matrix?

Yes. Augment the matrix A with the identity matrix I to form [A|I]. Apply Gauss-Jordan elimination to reduce A to I. The right half becomes A⁻¹. If the matrix is singular (determinant equals zero), the process will fail — no inverse exists.

What does it mean if I get a row of all zeros?

A row of all zeros (0 = 0) means one of your original equations is a linear combination of the others — it provides no new information. This results in a free variable and infinitely many solutions. However, if the row looks like [0 0 ... 0 | c] where c is non-zero, the system is inconsistent and has no solution.

What is Reduced Row Echelon Form (RREF)?

A matrix is in RREF when: (1) all leading entries (pivots) are 1, (2) each pivot is the only non-zero entry in its column, (3) pivots move to the right as you go down rows, and (4) any all-zero rows are at the bottom. RREF is unique for every matrix.

What is partial pivoting and why does it matter?

Partial pivoting means swapping rows so that the element with the largest absolute value in the current column becomes the pivot. This reduces numerical rounding errors that can accumulate during floating-point arithmetic, especially in larger matrices. Without partial pivoting, small pivot values can cause division by near-zero numbers, leading to inaccurate results.

What size matrices can this calculator handle?

This calculator supports matrices from 2×2 up to 6×6. For the Solve System and RREF modes, you enter an augmented matrix with n rows and n+1 columns. For the Inverse mode, you enter a square n×n matrix. All computations use partial pivoting for numerical stability.

How do I interpret the determinant shown in the results?

The determinant tells you about the nature of the system. If det ≠ 0, the system has a unique solution and the matrix is invertible. If det = 0, the coefficient matrix is singular — meaning the system either has infinitely many solutions or no solution at all. The determinant also scales areas (2D) or volumes (3D) under linear transformations.

Can I use Gauss-Jordan elimination for non-square systems?

Gauss-Jordan elimination works on any matrix, not just square ones. However, this calculator focuses on square systems (n equations in n unknowns), which is the most common use case. For non-square systems, RREF still reveals the rank and solution structure, but you may have free variables or no solution.

What is the difference between Gauss-Jordan and Cramer's Rule?

Cramer's Rule uses determinants of submatrices to find each variable individually. It works only for square systems with a unique solution (det ≠ 0). Gauss-Jordan elimination is more general — it handles singular systems, finds inverses, and is computationally more efficient for larger matrices (O(n³) vs O(n⁴) for Cramer's Rule).

Gauss-Jordan Elimination Calculator

Mode

Size

Output

Enter the augmented matrix [A|b] for your system of equations.

Solution

2.000000

3.000000

-1.000000

det = -1

rank = 3

Unique

Verification

Substituting back into original equations

2x + y - z = 8

LHS = 8, RHS = 8

Pass

- 3x - y + 2z = -11

LHS = -11, RHS = -11

Pass

- 2x + y + 2z = -3

LHS = -3, RHS = -3

Pass

RREF Matrix

Reduced Row Echelon Form

-1

Step-by-Step Solution

13 steps to reach the result

1Starting augmented matrix

-1

-3

-1

-11

-2

-3

2Swap R1 ↔ R2 (partial pivoting)

R1 ↔ R2

-3

-1

-11

-1

-2

-3

3Scale R1 by 1/-3 to make pivot = 1

R1 = R1 × (1/-3)

1/3

-2/3

11/3

-1

-2

-3

4Eliminate column 1 from row 2

R2 = R2 - (2) × R1

1/3

-2/3

11/3

1/3

2/3

-2

-3

5Eliminate column 1 from row 3

R3 = R3 - (-2) × R1

1/3

-2/3

11/3

1/3

2/3

5/3

2/3

13/3

6Swap R2 ↔ R3 (partial pivoting)

R2 ↔ R3

1/3

-2/3

11/3

5/3

2/3

13/3

1/3

2/3

7Scale R2 by 1/5/3 to make pivot = 1

R2 = R2 × (1/5/3)

1/3

-2/3

11/3

2/5

13/5

1/3

2/3

8Eliminate column 2 from row 1

R1 = R1 - (1/3) × R2

-4/5

14/5

2/5

13/5

1/3

2/3

9Eliminate column 2 from row 3

R3 = R3 - (1/3) × R2

-4/5

14/5

2/5

13/5

1/5

-1/5

10Scale R3 by 1/1/5 to make pivot = 1

R3 = R3 × (1/1/5)

-4/5

14/5

2/5

13/5

-1

11Eliminate column 3 from row 1

R1 = R1 - (-4/5) × R3

2/5

13/5

-1

12Eliminate column 3 from row 2

R2 = R2 - (2/5) × R3

-1

13Solution: x = 2, y = 3, z = -1

-1

What is Gauss-Jordan Elimination?

A systematic algorithm for solving linear systems in one pass

[A|b]

Augmented matrix

Coefficients + constants in one grid

RREF

Target form

Diagonal of 1s, zeros everywhere else

O(n³)

Complexity

Efficient for systems up to 6×6

Gauss-Jordan elimination extends Gaussian elimination by reducing an augmented matrix all the way to Reduced Row Echelon Form (RREF). Standard Gaussian elimination stops at an upper-triangular matrix and requires back-substitution; Gauss-Jordan eliminates entries both above and below each pivot, so the solution can be read directly from the final matrix.

Three use cases in one tool: solve a system of equations (Ax = b), reduce any matrix to RREF, or find the inverse A⁻¹ via [A|I] → [I|A⁻¹]. Switch modes in the input panel.

How does the algorithm work?

Three elementary row operations, applied systematically

Swap rows

R_i ↔ R_j

Partial pivoting for stability

Scale a row

R_i → (1/pivot) · R_i

Makes the pivot equal to 1

Row replacement

R_i → R_i − k · R_j

Zeros out above & below

Worked example — 3×3 system:

2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3
Form augmented matrix [2 1 −1 | 8; −3 −1 2 | −11; −2 1 2 | −3]
After Gauss-Jordan elimination → [1 0 0 | 2; 0 1 0 | 3; 0 0 1 | −1]
Solution: x = 2, y = 3, z = −1

The key difference from Gaussian elimination: after making each pivot equal to 1, you eliminate all other entries in that column — not just the ones below. This produces a diagonal of 1s with zeros everywhere else, so no back-substitution is needed.

Finding the inverse via Gauss-Jordan

Augment with the identity, reduce, and read off A⁻¹

Inverse transformation

[A | I] &xrarr; row operations &xrarr; [I | A⁻¹]

Only works when det(A) ≠ 0

Augment the n×n matrix A with the n×n identity matrix I to form a n×2n block [A|I]. Apply Gauss-Jordan elimination to the left half. If A is invertible, the left block becomes I and the right block becomes A⁻¹. If A is singular (det = 0), a row of zeros appears on the left — no inverse exists.

Cofactor method

O(n! × n²)

Gauss-Jordan method

O(n³)

Gauss-Jordan is far more efficient than cofactor expansion for matrices larger than 3×3, and naturally incorporates partial pivoting for numerical stability.

Common mistakes to avoid

Pitfalls that trip up students and practitioners

Skipping partial pivoting

Always swap in the row with the largest absolute value in the pivot column. Small pivots amplify rounding errors, especially in 5×5 and 6×6 matrices.

Only eliminating below the pivot

That produces Gaussian (upper triangular), not Gauss-Jordan (RREF). You must zero out entries above the pivot too.

Misreading a zero row as an error

A row [0 0 ... 0 | 0] means a dependent equation and infinitely many solutions — not a broken system.

Ignoring inconsistent rows

A row [0 0 ... 0 | c] where c ≠ 0 means 0 = c, which is impossible. The system has no solution.

Not verifying the solution

Plug the solution back into the original equations. Our calculator does this automatically in Solve mode.

Wrong augmented matrix dimensions

For solving Ax = b, the augmented matrix is n × (n+1). For inverse, it’s n × 2n. Mixing them up produces garbage.

Pro tips

Get the most out of this calculator

Use fraction mode for exact answers

Decimal output rounds; fraction mode preserves exact values like 1/3 or 7/11 that would otherwise lose precision.

Check the determinant first

If det = 0, the system is either dependent (infinite solutions) or inconsistent (no solution). The inverse doesn’t exist.

Share via URL

Every input is encoded in the URL. Copy the address bar to share an exact problem setup with a classmate or professor.

Frequently Asked Questions

Common questions about Gauss-Jordan elimination and RREF

Embed Gauss-Jordan Elimination Calculator

Add this calculator to your website or blog for free.

Related calculators from other categories

Last updated May 1, 2026

Gauss-Jordan Elimination Calculator

Verification

RREF Matrix

Step-by-Step Solution

What is Gauss-Jordan Elimination?

How does the algorithm work?

Finding the inverse via Gauss-Jordan

Common mistakes to avoid

Pro tips

Frequently Asked Questions

Embed Gauss-Jordan Elimination Calculator

You Might Also Like