Matrix Calculator — Determinant, Inverse, Multiply, RREF

This matrix calculator handles the operations from a first linear algebra course on matrices up to 10 × 10: addition, subtraction and multiplication, scalar multiples, transpose, determinant, inverse, rank, reduced row echelon form (RREF) and integer powers. Every step uses exact fraction arithmetic, so answers match the textbook — 3/5 rather than 0.6000000001 — with decimal versions alongside.

How to use this tool

1. Type matrix A with one row per line and entries separated by spaces or commas; fractions like 1/3 and decimals like 0.5 work.
2. For A + B, A − B or A × B, also fill in matrix B.
3. Choose the operation, plus k for a scalar multiple or n for a power.
4. Copy the result in bracket form [a, b; c, d] or read it from the tables.

Formula

(AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ · det [[a, b], [c, d]] = ad − bc · A⁻¹ = adj(A) ÷ det(A)

Matrices are added entry by entry and must have the same size; for A × B the number of columns of A must equal the rows of B. The determinant is found by row reduction (each row swap flips the sign), the inverse by reducing [A | I] to [I | A⁻¹], and the rank is the number of pivots in the reduced row echelon form. A matrix has an inverse only when its determinant is not zero. Each entry can be up to 30 characters long; if an exact answer would need numbers with more than 1,000 digits (a large matrix to a high power, say), the calculator says so instead of running for minutes.

Example

For A = [[2, 1], [1, 3]], det(A) = 2 × 3 − 1 × 1 = 5, so A is invertible and A⁻¹ = (1/5)[[3, −1], [−1, 2]] = [3/5, −1/5; −1/5, 2/5]. Multiplying [[1, 2], [3, 4]] by [[5, 6], [7, 8]] gives [[19, 22], [43, 50]].