|A| Determinant Calculator

Determinant Calculator

Calculate the determinant of any n×n matrix with full cofactor expansion steps, Laplace expansion, and LU decomposition verification.

|A|

Determinant Calculator

Enter your matrix, select size, then click Calculate

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Matrix A
Quick Examples

What Is a Determinant?

The determinant is a scalar value computed from a square matrix that encodes important properties of the matrix. It is written as det(A) or |A|.

Key Properties

  • det(A) ≠ 0 if and only if A is invertible (non-singular)
  • det(AB) = det(A) × det(B)
  • det(Aᵀ) = det(A)
  • det(kA) = kⁿ det(A) for an n×n matrix
  • Swapping two rows multiplies the determinant by −1

2×2 Determinant Formula

$$\det\begin{pmatrix}a & b \\ c & d\end{pmatrix} = ad - bc$$

3×3 Determinant (Sarrus Rule or Cofactor Expansion)

$$\det(A) = a_{11}(a_{22}a_{33}-a_{23}a_{32}) - a_{12}(a_{21}a_{33}-a_{23}a_{31}) + a_{13}(a_{21}a_{32}-a_{22}a_{31})$$

Frequently Asked Questions

A determinant of zero means the matrix is singular: its rows (or columns) are linearly dependent, it has no inverse, and the corresponding system of linear equations either has no solution or infinitely many solutions.
Cofactor expansion (Laplace expansion) expands the determinant along a row or column: det(A) = Σⱼ aᵢⱼ(-1)^(i+j) Mᵢⱼ, where Mᵢⱼ is the minor (determinant of the submatrix with row i and column j removed).
No. The determinant is only defined for square matrices (n×n). For non-square matrices you can compute the rank or use the pseudodeterminant, but standard determinants require equal rows and columns.