2×2 Matrix

For matrix [[a,b],[c,d]]: det = ad - bc

3×3 Matrix

Use Sarrus rule or cofactor expansion:

det = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)

4×4 Matrix

Use cofactor expansion along any row or column:

det = Σ(-1)^(i+j) × aᵢⱼ × det(Mᵢⱼ)

Properties

• If det = 0, matrix is singular (not invertible)
• If det ≠ 0, matrix is non-singular (invertible)
• det(AB) = det(A) × det(B)
• det(A^T) = det(A)