How do I add or subtract two matrices?

The matrices must have the same number of rows and columns. Add or subtract entries in matching positions.

What dimensions are required for matrix multiplication?

The number of columns in the first matrix must equal the number of rows in the second. The result has the first matrix's rows and the second matrix's columns.

Which matrices have a determinant?

A determinant is defined only for square matrices with equal row and column counts.

When does a matrix have an inverse?

A square matrix has an inverse when its determinant is nonzero. A matrix with determinant zero is singular and not invertible.

How do I transpose a matrix?

Turn rows into columns and columns into rows. Entry i,j of A moves to position j,i in the transpose.

What does matrix rank mean?

Rank is the number of linearly independent rows or columns and represents the dimension of independent information in the matrix.

How is matrix trace calculated?

Add the entries on the main diagonal running from the top left to the bottom right of a square matrix.

How are empty matrix cells handled?

After at least one cell is completed, any remaining empty cells are treated as zero.

Tools/🧮 Calculators/Matrix Calculator

Matrix Calculator

Add, subtract, and multiply matrices or calculate determinant, inverse, transpose, rank, and trace.

Version 1.0 · Last Updated: August 10, 2026 · Maintained by ToolLarder

Online matrix calculations and linear algebra operations

Enter matrices from 1×1 to 6×6 and perform common matrix operations in one place. The calculator checks dimension compatibility and displays results in your local number format.

Matrix addition and subtraction

Add or subtract corresponding entries of two matrices with equal row and column counts.

(A ± B)ᵢⱼ = aᵢⱼ ± bᵢⱼ

Matrix multiplication

When the columns of A equal the rows of B, each output entry is a row-column dot product.

(AB)ᵢⱼ = Σ aᵢₖbₖⱼ

Determinant calculator

Compute the determinant of a square matrix using elimination with pivoting; a zero determinant indicates a singular matrix.

det(A)

Inverse matrix calculator

Find the inverse of a nonsingular square matrix with Gauss-Jordan elimination.

A × A⁻¹ = I

Matrix transpose

Swap rows and columns to convert an m×n matrix into an n×m matrix.

(Aᵀ)ᵢⱼ = Aⱼᵢ

Matrix rank calculator

The number of nonzero rows in row-echelon form gives the rank of a matrix.

rank(A)

Matrix trace calculator

Add the entries on the main diagonal of a square matrix to find its trace.

tr(A) = Σ aᵢᵢ

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