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ALGEBRA / MATRICES

Matrix Calculator for Operations, Determinants, Inverses, RREF and Linear Systems

Updated By Unit Conversion Plus

Perform matrix operations using exact fractions. Calculate a determinant, inverse, transpose, reduced row echelon form or rank, multiply two matrices, or solve a linear system.

Enter your matrix

Matrix A size
Matrix B size

Matrix A

Matrix B

Enter integers, decimals or fractions such as −3, 0.25 or 2/5. Matrices can contain up to 5 rows and 5 columns.

What this matrix calculator can do

Use addition or subtraction for matrices with the same dimensions. For multiplication, the number of columns in A must equal the number of rows in B. Determinants and inverses apply only to square matrices. The transpose, RREF and rank operations also work with rectangular matrices.

Exact row reduction

The calculator performs Gaussian elimination using exact rational arithmetic. It shows row swaps, row scaling and row replacement operations, so values such as 1/3 remain fractions instead of becoming rounded decimals. RREF reveals pivot columns, rank and the structure of a system of equations.

Solving Ax = b

Select “Solve Ax = b”, enter a square coefficient matrix A and the constants in vector b. The calculator reduces the augmented matrix, identifies a unique solution where one exists, and verifies the answer by multiplying A by the computed vector. Singular systems are reported as having no unique solution.

Determinants and inverses

A zero determinant means a square matrix is singular and has no inverse. For an invertible matrix, row reduction transforms the augmented matrix [A | I] into [I | A⁻¹]. Multiplying A by the displayed inverse returns the identity matrix.

Frequently asked questions

Why can two matrices not always be multiplied?

If A has m × n entries and B has p × q entries, A × B exists only when n = p. The result then has m rows and q columns.

What is reduced row echelon form?

RREF is a standard matrix form in which every pivot equals 1, each pivot is the only nonzero entry in its column, and pivots move to the right as rows move downward.

What does matrix rank mean?

Rank is the number of pivot positions in the RREF. It measures how many rows or columns provide independent information.

Why does a matrix have no inverse?

A square matrix has an inverse only when its determinant is nonzero, which is equivalent to having full rank.