Determinant Solver

Type values — result updates instantly. Supports 2×2 to 6×6.

Size Decimals

Last edited cell: a[1,1]

What is a determinant?

It’s a scalar computed from a square matrix. Geometrically, it scales areas/volumes and encodes orientation. A zero determinant means the matrix is singular (non-invertible).

General formula

Leibniz formula: det(A) = Σ_σ sgn(σ) Π_i a_i,σ(i). Practical computation uses elimination/factorization for efficiency.

2×2, 3×3, 4×4

2×2: det([[a,b],[c,d]]) = ad − bc.

3×3: Use Sarrus’ rule or cofactor expansion.

4×4: Use cofactor expansion or LU; this calculator uses LU under the hood.

Properties

det(AB) = det(A)·det(B)
det(Aᵀ) = det(A)
Swap two rows → sign flips
Scale a row by k → determinant scales by k
Add multiple of a row to another → no change

Quick Breakdown

For 3×3, you’ll see Sarrus breakdown here; for other sizes, LU note appears.

How the Determinant Solver Works

This calculator parses your entries and computes the determinant in real time. For numerical stability and performance it uses LU decomposition with partial pivoting. The result equals the product of the diagonal of the U factor times the sign from row swaps.

Handles integers, decimals, and scientific notation (e.g., 1e-3).
Supports sizes 2×2 to 6×6. Larger sizes are possible but not exposed for mobile responsiveness.
Identity/Random buttons help you test quickly.
Adjust the Decimals control to format the display without changing the computed value.

Frequently Asked Questions

What is a determinant?

It’s a scalar computed from a square matrix. Geometrically, it scales areas/volumes and encodes orientation. A zero determinant means the matrix is singular (non-invertible).

General formula

Leibniz formula: det(A) = Σσ sgn(σ) Πi ai,σ(i). Practical computation uses elimination/factorization for efficiency.