About MatrixCalc
MatrixCalc is a free online matrix calculator. It performs the standard operations of linear algebra — addition, subtraction, multiplication, transpose, determinant, inverse, rank, trace, powers, eigenvalues, LU and Cholesky decomposition — on matrices up to 50×50, and shows the full solution steps for most of them, so you can follow the reasoning instead of just copying an answer.
Why it exists
Most matrix tools either hand you a number with no explanation, or bury the calculator under adverts and sign-up walls. This one runs entirely in your browser, needs no account, and is built to be genuinely useful for students checking homework and for anyone who needs a quick, reliable answer.
The explanation matters as much as the number. A calculator that says det(A) = −59 and stops has told you nothing you can use in an exam. Every operation page on this site therefore carries the mathematics behind the button: what the result means, how to reproduce it by hand, which method to pick for which matrix, and the specific mistakes that cost marks.
How the calculations work
Everything is computed locally in JavaScript — your matrices never leave your device. There is no maths library behind the scenes; the algorithms are implemented directly, and which one runs depends on the matrix:
- Determinants use the closed formula for 2×2 and 3×3, and LU factorisation with partial pivoting from 4×4 up. Cofactor expansion costs
n!operations and is unusable well before 50×50. - Inverses and rank come from Gauss-Jordan elimination on the augmented matrix.
- Eigenvalues use Jacobi rotations for symmetric matrices, which always converge, and the QR algorithm with a Wilkinson shift and deflation otherwise.
- The characteristic polynomial uses the Faddeev–LeVerrier recurrence, building the coefficients from traces of successive powers rather than expanding a symbolic determinant.
- The step-by-step panel runs a separate engine in exact rational arithmetic, which is why intermediate results show as fractions like 3/5 instead of 0.6000000001. A row that should reduce to exactly zero does, instead of leaving a residue like 1e−16 that would be miscounted as a pivot.
How the results are checked
The computation engine has an automated test suite covering arithmetic, elimination, decomposition, powers, eigenvalues and the matrix properties, and it runs on every change before anything reaches the site. Tests are how a bug gets fixed once rather than repeatedly: when a wrong answer is reported, the failing matrix becomes a test case first and the fix comes second.
That is also how the eigenvalue routine was corrected. An earlier version used an unshifted QR iteration that failed to converge on ordinary non-symmetric matrices — a triangular matrix with eigenvalues 2, 3 and 4 was enough to break it. The current version uses a Wilkinson shift with deflation, and the regression is pinned by tests.
Known limits
- Results are computed up to 50×50; step-by-step working stops at 8×8, above which it would run to hundreds of matrices and teach nothing.
- Eigenvalues are computed numerically and converge to real eigenvalues only. A matrix with a complex spectrum — a rotation, for instance — reports that it did not converge rather than inventing an answer.
- Eigenvalues and eigenvectors have no step-by-step panel. The algorithms are iterative rather than algebraic, so their intermediate values are rounding artefacts, not reasoning you could reproduce by hand.
- The numeric result path uses floating point; only the step-by-step engine is exact.
Languages
The calculator interface and the written explanations are available in ten languages: English, Portuguese, Spanish, French, German, Russian, Indonesian, Chinese, Japanese and Arabic. Each language has its own URLs and its own text — no page is a machine translation of another shown under a different flag. Your choice is remembered in your browser.
Contact
Found a bug, got a wrong answer, or want an operation added? Email franfactory.store@gmail.com. If you are reporting a wrong result, please include the matrix and the operation you ran — it makes the fix much faster, and it is what turns your report into a permanent test case.