Matrix Calculator

Perform matrix operations including addition, multiplication, determinant, inverse, transpose, and eigenvalues.

How to Use Matrix Calculator

1

Enter matrix values

Fill in the elements for one or two matrices.
2

Select operation

Choose addition, multiplication, determinant, inverse, or transpose.
3

View the result

See the calculated result matrix or scalar value.

Tested tool guide Tested browser tools Checked August 16, 2026

What Matrix Calculator does, with a checked example

This calculator evaluates one matrix or a pair of matrices using the selected operation. It covers addition, matrix multiplication, determinant, inverse, transpose, and eigenvalues. Matrix dimensions determine which operations are defined: addition needs matching shapes, while multiplication needs matching inner dimensions. The common surprise is that matrix multiplication is row-by-column, not cell-by-cell. Reversing the operands can change the result, change its dimensions, or make the product undefined.

Worked example

A concrete input and expected output from the current implementation.

Input

Operation: A x B
A = [[1, 2], [3, 4]]
B = [[2, 0], [1, 2]]

Expected output

[[4, 4], [10, 8]]

The first entry is 1 x 2 + 2 x 1 = 4. The remaining row-column products are 1 x 0 + 2 x 2 = 4, 3 x 2 + 4 x 1 = 10, and 3 x 0 + 4 x 2 = 8.

How the result is produced

Entry and dimension rules

For addition, corresponding entries are combined, so both matrices must have the same dimensions. For multiplication, each output entry is the dot product of one row from the left matrix and one column from the right matrix. The left matrix's column count must equal the right matrix's row count. Transposition exchanges row and column indices.

Square-matrix operations

Determinant, inverse, and eigenvalue calculations apply to square matrices. The determinant is a scalar, and a zero determinant means no inverse exists. An inverse, when it exists, produces the identity matrix when multiplied by the original. Eigenvalues are scalars lambda for which the matrix A - lambda I is singular.

Good uses

Multiply transformation matrices in a specified order and verify the resulting coordinates or coefficients.
Check the determinant of a coefficient matrix before attempting to solve a small linear system with its inverse.
Find eigenvalues when studying the characteristic behavior, repeated modes, or long-term action of a square matrix.

Limits and checks

A dimension mismatch is a mathematical restriction, not merely an input-format error. Same-sized matrices can be added, but AB requires the columns of A to equal the rows of B.
A zero determinant makes the inverse undefined. With rounded decimal entries, a very small nonzero determinant can resemble zero, and the inverse may be highly sensitive to small input changes.
Do not treat eigenvalues as eigenvectors or as proof that a matrix is diagonalizable. A repeated eigenvalue can have fewer independent eigenvectors than its multiplicity, and eigenvalue order has no mathematical significance.

Common questions

Why does A x B differ from B x A?

Each entry of AB pairs a row of A with a column of B, while BA pairs a row of B with a column of A. These are different dot products and can produce matrices with different shapes. Equality occurs for special pairs of matrices that commute, but it is not a general property of matrix multiplication.

Can every square matrix be inverted?

No. A square matrix is invertible only when its determinant is nonzero. If its rows or columns are linearly dependent, its determinant is zero and no inverse exists. For decimal data close to singular, an inverse can contain very large values, so multiply the reported inverse by the original matrix and check whether the product is close to the identity.

References and verification

The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.