Matrix Calculator

Type your matrices and get exact answers with the working.

Private by design

Your inputs stay on your device and are not sent to our servers. Optional usage analytics does not receive your files or tool inputs.

Matrix Calculator

Exact arithmetic, inverse, determinant, rank and eigenvalues

Operation

Matrices

Matrix AOne row per line; separate numbers with spaces or commas. Fractions like 1/2 are fine. Up to 6 × 6.

Matrix B

A × B2 × 2 matrix

A × B (2 × 2)

19	22
43	50

How it works

A matrix is a grid of numbers. Multiplying matrices combines rows with columns, the determinant measures how a square matrix scales area or volume, the inverse undoes a matrix, the rank counts independent rows, and eigenvectors are directions that a matrix only stretches.

A is 2 × 2 and B is 2 × 2, so A × B is 2 × 2.
Each entry is a row of A times a column of B: (A×B)ᵢⱼ = Σ aᵢₖ × bₖⱼ.
Row 1 · column 1: 1×5 + 2×7 = 19
Row 1 · column 2: 1×6 + 2×8 = 22
Row 2 · column 1: 3×5 + 4×7 = 43
Row 2 · column 2: 3×6 + 4×8 = 50

Matrix multiplication needs the columns of A to equal the rows of B, and A × B is usually not B × A.
A matrix has an inverse only when its determinant is not 0.
The sum of the eigenvalues is the trace and their product is the determinant, a good check.

Entries are read as exact fractions, so determinant, inverse, rank, powers and rational eigenvalues have no rounding. Determinant: 2×2 formula, cofactor expansion for 3×3 and row reduction beyond. Inverse and rank: Gauss–Jordan elimination. Eigenvalues: the exact characteristic polynomial (Faddeev–LeVerrier), exact rational roots by the rational root theorem, remaining roots by numerical iteration to about 10 digits.

Preparing tool

How to use Matrix Calculator

Type matrix A with one row per line and numbers separated by spaces (fractions like 1/2 work).
Pick the operation. Add matrix B when the operation needs two.
Read the exact answer and follow the row-by-row working.

Read the detailed guide →

What are the limits of Matrix Calculator?

Matrices up to 6 × 6 with real entries. Determinant, inverse, rank and powers are exact (fractions). Eigenvalues of irrational or complex values are numerical (about 10 digits); matrices of degree 6 may show small rounding. Nothing you enter leaves your device.