📊 Cofactor Expansion Calculator
Calculate determinant using cofactor expansion (Laplace expansion)
How to Use This Calculator
Select Matrix Size
Choose 2×2, 3×3, or 4×4 matrix size.
Enter Matrix Elements
Input all elements of your square matrix.
Choose Expansion Row
Select which row to expand along (any row gives the same determinant).
View Expansion
See step-by-step cofactor expansion showing each term and the final determinant.
Formula
det(A) = Σⱼ aᵢⱼ × Cᵢⱼ (expansion along row i)
Cᵢⱼ = (-1)ⁱ⁺ʲ × det(Mᵢⱼ) where Mᵢⱼ is the minor matrix
Cofactor Expansion (Laplace Expansion):
For any row i: det(A) = Σⱼ₌₁ⁿ aᵢⱼ × Cᵢⱼ
For any column j: det(A) = Σᵢ₌₁ⁿ aᵢⱼ × Cᵢⱼ
Cofactor:
Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ
where Mᵢⱼ = determinant of the (n-1)×(n-1) matrix obtained by removing row i and column j
Example: 3×3 Matrix
Expanding along row 1:
det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃
= a₁₁ × det([a₂₂ a₂₃; a₃₂ a₃₃]) - a₁₂ × det([a₂₁ a₂₃; a₃₁ a₃₃]) + a₁₃ × det([a₂₁ a₂₂; a₃₁ a₃₂])
About Cofactor Expansion Calculator
The Cofactor Expansion Calculator computes the determinant of a square matrix using cofactor expansion (also called Laplace expansion). You can expand along any row or column - the result is always the same. This method is fundamental for understanding determinants.
When to Use This Calculator
- Linear Algebra: Calculate determinants step-by-step
- Education: Learn how cofactor expansion works
- Matrix Theory: Understand determinant computation
- Verification: Check determinant calculations manually
Why Use Our Calculator?
- ✅ Step-by-Step: Shows each term in the expansion
- ✅ Flexible: Expand along any row
- ✅ Detailed: Displays minors and cofactors
- ✅ Educational: Helps understand Laplace expansion
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Minor: The determinant of the submatrix after removing one row and one column
- Cofactor: Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ, includes the sign
- Expansion: det(A) = Σ aᵢⱼ × Cᵢⱼ for any row i or column j
- Flexibility: You can expand along any row or column - result is the same
- Recursive: Process continues until reaching 2×2 or 1×1 determinants
Tips
- Choose a row or column with many zeros to simplify calculations
- Each term alternates signs: +, -, +, -, ...
- Expanding along a row with zeros reduces computation
Frequently Asked Questions
What is cofactor expansion?
Cofactor expansion (Laplace expansion) is a method to compute determinants by expanding along a row or column: det(A) = Σ aᵢⱼ × Cᵢⱼ, where Cᵢⱼ are cofactors.
Does it matter which row or column I expand along?
No! Expanding along any row or column gives the same determinant. However, choosing a row/column with many zeros makes calculation easier.
What's the difference between minor and cofactor?
A minor Mᵢⱼ is the determinant of the submatrix. A cofactor Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ includes the sign factor (-1)ⁱ⁺ʲ.
How do I know the sign for each cofactor?
The sign pattern follows a checkerboard: Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ. Starting from position (1,1) with +, it alternates: +, -, +, -, ...
Is cofactor expansion efficient?
For small matrices (≤4×4), it's fine. For larger matrices, other methods like Gaussian elimination are faster, but cofactor expansion is educational and useful for symbolic computation.