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Determinant Solver

Determinant Solver

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

Last edited cell: a[1,1]

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.

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.

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

Matrix: [[0, 0, 0], [0, 0, 0], [0, 0, 0]]
Sarrus: (a·e·i + b·f·g + c·d·h) − (c·e·g + a·f·h + b·d·i)
= (0) − (0)
= 0