# Matrix & Linear Algebra Lab

> Exact-fraction RREF with steps, determinant, inverse, null space, eigenvalues, LU/QR and Ax = b. Enter matrices with whole numbers, fractions or decimals and get exact results: row reduction with every row operation, determinant, inverse, rank, null space, A + B, A·B and solutions of A x = b. Eigenvalues are exact for 2×2/3×3 cases with rational or quadratic roots and numeric (shifted QR algorithm) otherwise; LU, QR and the condition number are computed numerically.

Part of usefulweb.site (https://usefulweb.site/). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://matrixlinearalgebralab.usefulweb.site/ (interactive tool at https://matrixlinearalgebralab.usefulweb.site/tool)
- JSON API: POST https://matrixlinearalgebralab.usefulweb.site/api/v1/calculate with a JSON object of inputs (schema: https://matrixlinearalgebralab.usefulweb.site/api/v1/tool)
- OpenAPI: https://matrixlinearalgebralab.usefulweb.site/openapi.json
- MCP (Streamable HTTP): https://matrixlinearalgebralab.usefulweb.site/mcp - tool `matrix_linear_algebra_lab`

## Inputs
- operation (select): Operation
- matrixA (textarea, required): Matrix A - One row per line (or rows separated by ;), entries separated by spaces or commas. Fractions like 1/3 and decimals are fine. Up to 10×10.
- matrixB (textarea): Matrix B - Only for A + B, A − B and A · B.
- vectorB (text): Right-hand side b - Only for "Solve A x = b": one number per row of A.

## Method
- Entries are parsed as exact fractions (BigInt numerators and denominators), so row reduction, determinants, inverses, ranks, null spaces and solutions of A x = b are exact, with no rounding.
- Row reduction is Gauss–Jordan elimination: swap to a non-zero pivot, scale the pivot row to 1, clear the column above and below; every operation is listed with the matrix after it.
- Eigenvalues: the characteristic polynomial is computed exactly (Faddeev–LeVerrier). Rational roots and a remaining quadratic give exact eigenvalues (with surds or complex values); otherwise the matrix is reduced to Hessenberg form and the shifted QR algorithm finds the eigenvalues numerically. Eigenvectors are exact null spaces for rational eigenvalues and numeric (inverse iteration) otherwise.
- LU uses partial pivoting (P·A = L·U); QR uses Householder reflections; the condition number is σmax/σmin from the singular values (Jacobi eigenvalues of AᵀA). These are double-precision numeric results.