Gaussian Elimination Solver

Name: Gaussian Elimination Solver
Availability: InStock
Rating: 4.7 (852 reviews)
Author: Generatr

Row echelon Ax=b

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4.7(852 reviews)

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Gauss Elim

Row echelon + back sub

System

rank = 3 / 3

Solution x

(2.000000, 3.000000, -1.000000)

Row Operations

Augmented [A|b]

2.000  1.000  -1.000  8.000
-3.000  -1.000  2.000  -11.000
-2.000  1.000  2.000  -3.000

Swap R1 ↔ R2

-3.000  -1.000  2.000  -11.000
2.000  1.000  -1.000  8.000
-2.000  1.000  2.000  -3.000

R2 -= -0.667·R1

-3.000  -1.000  2.000  -11.000
0.000  0.333  0.333  0.667
-2.000  1.000  2.000  -3.000

R3 -= 0.667·R1

-3.000  -1.000  2.000  -11.000
0.000  0.333  0.333  0.667
0.000  1.667  0.667  4.333

Swap R2 ↔ R3

-3.000  -1.000  2.000  -11.000
0.000  1.667  0.667  4.333
0.000  0.333  0.333  0.667

R3 -= 0.200·R2

-3.000  -1.000  2.000  -11.000
0.000  1.667  0.667  4.333
0.000  0.000  0.200  -0.200

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About Gaussian Elimination Solver

A Gaussian elimination solver for linear systems Ax=b. Performs row operations to reach row echelon form, then back-substitutes. Shows each row operation step-by-step. Detects rank, inconsistency, and free variables. Select from preset systems. All calculations are client-side.

Gaussian Elimination Solver Features

Row echelon
Back sub
Step display
Rank
Presets

Gaussian elimination: row reduce [A|b] to upper triangular form using elementary row operations (swap, scale, add multiple). Then back-substitute. O(n³/3) operations. The fundamental direct method for linear systems.

How to Use

Set up the system:

[A|b]: Augmented matrix
Steps: Row operations shown
Output: REF + solution

Partial Pivoting

Swap rows to put largest element in pivot position. Prevents division by small numbers (numerical instability). Always used in practice.

Complexity

O(n³/3) multiplications for elimination. O(n²/2) for back-substitution. Total: O(n³/3). LU decomposition is equivalent but stores the work for reuse.

Step-by-Step Instructions

1Select a system.
2View row operations.
3See echelon form.
4Back-substitute.
5Get solution.

Gaussian Elimination Solver — Frequently Asked Questions

What are elementary row operations?+

Three types: (1) Swap two rows, (2) Multiply a row by non-zero scalar, (3) Add a multiple of one row to another. Any matrix can be reduced to row echelon form using these operations.

When does the system have no solution?+

When a row becomes [0 0 ... 0 | c] with c≠0. This means 0=c, a contradiction. The system is inconsistent — the equations are contradictory.

What are free variables?+

When rank(A) < n (fewer pivots than unknowns), some variables are 'free' — they can take any value. The solution forms a vector space. Each free variable adds one dimension to the solution space.

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