Determinant Calculator
Calculate the determinant of 2×2, 3×3, and 4×4 matrices with step-by-step cofactor expansion
det(A) = Σ (-1)^(i+j) × a_ij × M_ij
Calculate the determinant using cofactor expansion method
Quick Examples
Error
Determinant of Matrix A
Step-by-Step Solution (2×2)
Apply the 2×2 formula
det(A) = a₁₁ × a₂₂ - a₁₂ × a₂₁
Substitute values
Calculate result
Cofactor Expansion (First Row)
Minor: , Cofactor:
What this means
A determinant of 0 means the matrix is singular and has no inverse. The rows/columns are linearly dependent.
A non-zero determinant means the matrix is invertible. The absolute value || represents the scaling factor for area/volume transformations.
Determinant Formulas
| Size | Formula | Method |
|---|---|---|
| 2×2 | ad - bc | Direct formula |
| 3×3 | Σ aᵢⱼ × Cᵢⱼ | Cofactor expansion |
| 4×4 | Σ aᵢⱼ × Cᵢⱼ | Recursive cofactors |
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About Determinant Calculator
What is a Determinant?
The determinant is a scalar value that can be computed from a square matrix. It provides important information about the matrix:
- A non-zero determinant means the matrix is invertible
- A zero determinant means the matrix is singular (not invertible)
- The absolute value represents the scaling factor for area/volume transformations
Determinant Formulas
2×2 Matrix
For a 2×2 matrix:
A = | a b |
| c d |
det(A) = ad - bc
3×3 Matrix (Sarrus' Rule or Cofactor Expansion)
For a 3×3 matrix, using cofactor expansion along the first row:
det(A) = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)
4×4 Matrix (Cofactor Expansion)
For larger matrices, we use recursive cofactor expansion, breaking it down into smaller determinants.
Properties of Determinants
- det(AB) = det(A) × det(B)
- det(Aᵀ) = det(A) (transpose has same determinant)
- det(kA) = kⁿ × det(A) for n×n matrix
- Swapping two rows/columns negates the determinant
- A row/column of zeros means det = 0
Applications
- Solving linear equations (Cramer's Rule)
- Matrix invertibility testing
- Eigenvalue calculations
- Area and volume calculations
- Change of variables in calculus (Jacobian)