Matrix Calculator
Add, multiply, transpose, invert and reduce matrices typed a row per line, with the determinant, rank and the solution of Ax = b from one elimination.
The result matrix or the scalar, with the determinant, rank and trace where they apply, the reduced row echelon form, and a warning when the matrix is singular or too ill-conditioned to trust.
Example: [1 2; 3 4] × [5 6; 7 8] = [19 22; 43 50]; the inverse of [4 7; 2 6] is [0.6 −0.7; −0.2 0.4] and its determinant is 10.
One elimination,
most of linear algebra.
How a matrix is read, what a single Gauss-Jordan pass gives you, and where floating point stops being honest.
Reading and multiplying
A matrix is typed a row per line, entries separated by spaces, commas or a slash for a fraction; every row has to be the same length, and the page says which row broke that rule rather than guessing. Addition and subtraction work entry by entry and need identical shapes. Multiplication is the row-by-column sum and needs the columns of A to match the rows of B — it is also not commutative, so A × B and B × A are different products and often only one of them exists. The element-by-element product is a separate choice here because it is the one people reach for by accident.
What one pass gives you
Gauss-Jordan elimination with partial pivoting — always taking the largest remaining entry in a column as the pivot, which is what keeps the arithmetic stable — produces the determinant, the rank, the reduced row echelon form, the inverse (by reducing A alongside the identity) and the solution of Ax = b, all from the same routine. That is why those answers agree with each other: a zero determinant, a rank below full and a missing inverse are three ways of saying the same thing. When a system has more unknowns than pivots the page names the free columns and shows one solution with them set to zero, saying plainly that it is a solution and not the solution.
Limits
Double precision, with a pivot tolerance you can see. An entry that should be exactly zero can arrive as 1e-17 and is rounded away for display, so a determinant reported as zero means “below the tolerance”, not “provably zero” — exact rational and symbolic arithmetic are not done here. Ill-conditioned matrices are flagged by the spread of their pivots, because such a matrix can be inverted and still give an answer with no correct digits. No eigenvalues, no LU, QR or SVD decomposition, no pseudo-inverse and no least-squares fit. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Gauss-Jordan elimination with partial pivoting gives the determinant, rank, inverse, echelon form and the solution of Ax = b; double precision with a stated pivot tolerance, not exact arithmetic
Last reviewed 22 September 2026. How results are checked: How we verify.