Gauss Elimination Calculator
Verified Calculation Engine
The online Gauss Elimination Calculator helps you calculate instantly and solve problems related to Matrices Determinants. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Engineering Math calculations.
Solve A x = b by Gaussian elimination. Enter Coefficient Matrix A and Constants Vector b to obtain a unique solution, infinitely many solutions, or no solution — with row-operation steps.
Gaussian elimination on [A | b] → row-echelon form → back-substitution / classification
Form the augmented matrix [A | b], perform row swaps and elimination to produce row-echelon form, then classify: unique solution (full rank), infinitely many solutions (rank < n, consistent), or no solution (inconsistent row).
- A is square n×n; b has length n.
- Entries are real numbers.
- Partial pivoting is used when the leading pivot is zero or small.
Inputs
Results
Worked Examples
2×2 unique solution
2x+3y=8, x+2y=5 → x=1, y=2.
- matrix: [ [ 2, 3 ], [ 1, 2 ] ]
- vector: [ [ 8 ], [ 5 ] ]
- classification: UNIQUE SOLUTION
- result: x = 1, y = 2
3×3 unique solution
System with solution (1,2,3).
- matrix: [ [ 1, 1, 1 ], [ 2, -1, 1 ], [ 1, 2, -1 ] ]
- vector: [ [ 6 ], [ 3 ], [ 2 ] ]
- classification: UNIQUE SOLUTION
- result: x = 1, y = 2, z = 3
Inconsistent system (no solution)
x+y=2 and 2x+2y=5 are inconsistent.
- matrix: [ [ 1, 1 ], [ 2, 2 ] ]
- vector: [ [ 2 ], [ 5 ] ]
- classification: NO SOLUTION
- result: NO SOLUTION
About this calculator
Overview
Gaussian elimination solves the linear system A x = b by transforming the augmented matrix [A | b] into row-echelon form with elementary row operations, then classifying the system as unique, infinite, or inconsistent.
When to use
Use this calculator to solve small linear systems by hand-style elimination, to verify homework pivots and row swaps, or to see why a system has no solution or free variables.
Inputs explained
- Coefficient Matrix A — square n×n
- Constants Vector b — column vector of length n
Formula / method
Build [A | b]. Use partial pivoting, then eliminate below each pivot (Ri ← Ri − k Rj). Read the row-echelon form: a row [0 … 0 | c] with c ≠ 0 means no solution; fewer pivots than unknowns with consistency means infinitely many solutions; a full set of pivots means a unique solution via back-substitution.
Worked example
Example: 2x+3y=8 and x+2y=5 gives UNIQUE SOLUTION x=1, y=2.
Interpreting results
Classification is primary. For a unique solution, unknowns are listed. For infinite solutions, a parametric form is shown. For no solution, the inconsistent row is identified in the steps.
Assumptions
- A is square; b matches A’s row count.
- Floating-point pivots use a small tolerance.
Limitations
- This is not a full CAS; large ill-conditioned systems need careful interpretation.
- Unlike Matrix Rank alone, this tool classifies the full linear system A x = b.
- Critical engineering decisions should be cross-checked by hand or with certified software.
Important note
Educational engineering-mathematics tool. Verify critical designs with hand analysis or certified software.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
Build [A | b]. Use partial pivoting, then eliminate below each pivot (Ri ← Ri − k Rj). Read the row-echelon form: a row [0 … 0 | c] with c ≠ 0 means no solution; fewer pivots than unknowns with consistency means infinitely many solutions; a full set of pivots means a unique solution via back-substitution.
Example Calculation
Example: 2x+3y=8 and x+2y=5 gives UNIQUE SOLUTION x=1, y=2.
Frequently Asked Questions
What is Gauss Elimination Calculator?
- Gaussian elimination solves the linear system A x = b by transforming the augmented matrix [A | b] into row-echelon form with elementary row operations, then classifying the system as unique, infinite, or inconsistent.
How does Gauss Elimination Calculator work?
- Build [A | b]. Use partial pivoting, then eliminate below each pivot (Ri ← Ri − k Rj). Read the row-echelon form: a row [0 … 0 | c] with c ≠ 0 means no solution; fewer pivots than unknowns with consistency means infinitely many solutions; a full set of pivots means a unique solution via back-substitution.
Why use this Engineering Math calculator?
- Use this calculator to solve small linear systems by hand-style elimination, to verify homework pivots and row swaps, or to see why a system has no solution or free variables.