🔢Matrix Calculator

About this tool

Type matrix A and matrix B (or a column vector b) one row per line, then choose an operation: A + B, A − B, A × B, determinant, inverse, transpose, rank with reduced row echelon form, power, or solve Ax = b. Results show as exact fractions or decimals, and the solver says when a system has no solution or infinitely many.

Exact answers and how they're found

Most online matrix tools work in floating point, so the inverse of a simple 3 × 3 comes back as 0.333333 and −1.110223e-16 instead of 1/3 and 0. This calculator uses exact rational arithmetic, so inputs given as integers, decimals or fractions like 3/4 stay exact through every step. Everything rests on Gauss–Jordan elimination — swap rows to bring a non-zero pivot up, scale it to 1, clear the rest of its column — the method taught in every linear algebra course. The determinant is the product of the pivots (with a sign change for each row swap), the rank is the number of pivots, the inverse comes from reducing [A | I] to [I | A⁻¹], and solving Ax = b reduces [A | b]: a row of zeros equal to a non-zero number means no solution, and fewer pivots than unknowns means infinitely many. Because the reduced row echelon form is shown, you can check homework steps against it. Honest notes: long decimals turn into long fractions — switch the display to decimals — and eigenvalues are not included. The fraction calculator handles single fractions, the quadratic solver one-variable equations, the scientific calculator general sums, and Fractions, decimals and percentages: the same number three ways, and how to convert explains the conversions.

Frequently asked questions