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Gaussian-elimination solution for small Ax = b systems with determinant and residual table.
Linear System Solver applies its registered numerical method in the browser after you select Solve Linear System. It reports solution vector, determinant, and residual table and hides prior output when a relevant input changes.
Configure inputs first. Results, visuals, downloads, and copy summary remain hidden until Solve Linear System succeeds.
Expressions use a restricted math parser and run in the browser. The tool accepts finite real-number inputs only and does not execute arbitrary code.
A linear system represents simultaneous equations in matrix form A x = b. Elimination and factorization methods transform the coefficient matrix to recover unknowns, while singularity, conditioning, scaling, pivoting, and floating-point precision determine whether a numerical solution is stable and meaningful.
Linear System Solver applies this concept to its defined inputs and workflow. It parses a coefficient matrix and right-hand-side vector, applies partial-pivot Gaussian elimination, and reports solution components with residual checks.
The calculator evaluates matrix table and vector table and returns solution vector, determinant, and residual table using the stated method. The selected units and sign or reference conventions must describe one consistent physical case. Users should interpret the output within the assumptions shown on the page and repeat the check with trusted reference values when the result affects engineering or research decisions. Matrix size is intentionally limited.
The Linear System Solver parses a coefficient matrix and right-hand-side vector, applies partial-pivot Gaussian elimination, and reports solution components with residual checks.
The solver applies pivoted elimination to Ax=b and reports the solution, residual vector, pivot information, and singularity or scaling warnings.
The successful result separates solution vector, determinant, and residual table so the estimate, diagnostic history, and visual evidence can be checked together.
Gaussian elimination transforms the coefficient matrix into upper-triangular form with partial pivoting, then uses back substitution to compute the solution vector.
A x = bA is the coefficient matrix, x is the unknown vector, and b is the right-hand side.
r = A x - bResiduals check how closely the solution satisfies the original equations.
| Symbol | Meaning | Unit |
|---|---|---|
| A | Coefficient matrix | problem units |
| x | Unknown solution vector | varies |
| b | Right-hand-side vector | varies |
| r | Residual vector | same as b |
This numerical-method tool is intended for educational, estimation, and preliminary engineering or research use. Always verify critical numerical results with analytical checks where possible, validated software, applicable standards, measured data, and qualified professional judgment before using results in real design, manufacturing, construction, safety, research, or compliance decisions.
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