usefulweb.site Matrix & Linear Algebra Lab

Math & science

Matrix & Linear Algebra Lab

Exact-fraction RREF with steps, determinant, inverse, null space, eigenvalues, LU/QR and Ax = b.

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.

Only for A + B, A − B and A · B.

Only for "Solve A x = b": one number per row of A.

Try an example: Overview of a 3×3 matrix · Solve A x = b with steps · Inverse with fractions · Eigenvalues of a 2×2 matrix · Null space of a rank-deficient matrix · Product A·B · QR decomposition

How it works

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.

  • 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.

For scripts and AI agents

  • JSON API: POST https://matrixlinearalgebralab.usefulweb.site/api/v1/calculate (same calculation as this page)
  • Interface description: OpenAPI 3.1 · AI guide: llms.txt
  • MCP server (Streamable HTTP): https://matrixlinearalgebralab.usefulweb.site/mcp · tool name matrix_linear_algebra_lab
Example request
curl -s -X POST https://matrixlinearalgebralab.usefulweb.site/api/v1/calculate \
  -H 'content-type: application/json' \
  -d '{"operation":"analyze","matrixA":"2 0 0\n0 3 4\n0 4 9"}'

Related tools

  • Step-by-Step Equation Solver – Solve linear, quadratic, rational and radical equations or linear systems with every step shown.
  • Math Workbench – Simplify, factor, differentiate, integrate, solve, take limits, series and plots in one box.
Feedback