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Matrix Determinant Calculator

Computes the determinant of a 2×2, 3×3 or 4×4 matrix by Laplace (cofactor) expansion along the first row, and reports whether the matrix is singular, its trace and its rank.

When to use

You need the determinant of a small square matrix, want to check whether it is invertible, or need its rank or trace.

Do not use when: You need the inverse matrix itself (use matrix-inverse), want to solve equations (use linear-system), or the matrix is larger than 4×4.

Formula

2×2: det = a·d − b·c; n×n: det = Σ_j (−1)^(1+j) · a_1j · M_1j, where M_1j is the determinant of the matrix without row 1 and column j (Laplace expansion)

Inputs

Parameter	Type	Unit	Required	Description
`matrix`	number_list		yes	All entries row by row: 4 values for 2×2, 9 for 3×3, 16 for 4×4. [[1, 2], [3, 4]] is entered as 1, 2, 3, 4.
`size`	integer		no	Matrix dimension n (2, 3 or 4). Optional: inferred from the number of entries when omitted. Range: ≥ 2, ≤ 4

Outputs

Output	Type	Description
`determinant`	number	det(A) by cofactor expansion along the first row.
`is_singular`	boolean	true when the determinant is 0 (the matrix has no inverse).
`expansion`	string	The first-row expansion with the numbers filled in, e.g. '1·4 − 2·3 = -2'; for 3×3 and 4×4 each term shows the entry times its minor.
`trace`	number	Sum of the diagonal entries.
`rank`	integer	Number of linearly independent rows (Gaussian elimination).
`size`	integer	Dimension n of the matrix.

Example

[[1, 2], [3, 4]]: {"matrix":[1,2,3,4],"size":2} → {"determinant":-2,"is_singular":false,"expansion":"1·4 − 2·3 = -2","trace":5,"rank":2,"size":2}

[[6, 1, 1], [4, -2, 5], [2, 8, 7]]: {"matrix":[6,1,1,4,-2,5,2,8,7]} → {"determinant":-306,"is_singular":false,"expansion":"6·(-54) − 1·(18) + 1·(36) = -306","trace":11,"rank":3,"size":3}

GET https://tttkmbb.com/api/v1/calculate/matrix-determinant?matrix=1%2C2%2C3%2C4&size=2

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/matrix-determinant (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/matrix-determinant · Markdown: https://tttkmbb.com/math/matrix-determinant.md · JSON definition: https://tttkmbb.com/math/matrix-determinant.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="matrix-determinant"
OpenAPI operationId: calculate_matrix_determinant
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Determinant (reference)
Wikipedia – Laplace expansion (reference)
Wolfram MathWorld – Determinant (reference)

FAQ

What does a zero determinant mean?

The rows (and columns) are linearly dependent: the matrix has no inverse, and the linear system it represents has no unique solution. The rank output shows how many independent rows remain.

Is the result exact?

For integer entries the cofactor expansion uses only integer multiplications and additions, so the determinant is exact as long as intermediate products stay below 2^53.

Related calculators

Matrix Inverse Calculator — Inverse and adjugate of the same matrix.
System of Linear Equations Solver — Cramer's rule uses these determinants to solve equations.
Vector Calculator — The 3D cross product is a 3×3 determinant.