🔢 Cofactor Matrix Calculator
Calculate the cofactor matrix and related matrices
How to Use This Calculator
Select Matrix Size
Choose 2×2, 3×3, or 4×4 for your square matrix.
Enter Matrix Elements
Input all elements of your square matrix.
Calculate
Click to compute the cofactor matrix, minor matrix, adjoint, and inverse.
Review Results
Compare minor matrix (without signs) and cofactor matrix (with signs).
Formula
Cofactor: Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ
Where Mᵢⱼ is the minor (determinant of submatrix after removing row i and column j)
Minor Matrix:
Mᵢⱼ = determinant of the (n-1)×(n-1) matrix obtained by removing row i and column j from A
Cofactor Matrix:
C = [Cᵢⱼ] where Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ
The sign pattern follows: +, -, +, -, ... (checkerboard pattern)
Adjoint Matrix:
adj(A) = Cᵀ (transpose of cofactor matrix)
Inverse:
A⁻¹ = adj(A) / det(A) (if det(A) ≠ 0)
About Cofactor Matrix Calculator
The Cofactor Matrix Calculator computes the cofactor matrix of a square matrix. The cofactor matrix contains cofactors Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ, where Mᵢⱼ are minors. The transpose of the cofactor matrix is the adjoint, which is used to find the inverse matrix.
When to Use This Calculator
- Matrix Inversion: Find inverse using adjoint method
- Linear Algebra: Understand cofactors and minors
- Determinant Calculation: Use cofactor expansion
- Education: Learn matrix theory concepts
Why Use Our Calculator?
- ✅ Complete Solution: Shows minor, cofactor, adjoint, and inverse
- ✅ Visual Comparison: Compare minor vs cofactor matrices
- ✅ Step-by-Step: See how cofactors relate to minors
- ✅ Educational: Understand the sign pattern
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Minor: Mᵢⱼ = determinant of submatrix after removing row i and column j (no sign)
- Cofactor: Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ (includes sign based on position)
- Sign Pattern: Starting from (1,1) with +, alternates: +, -, +, -, ...
- Adjoint: Transpose of cofactor matrix, used for finding inverse
- Relationship: A × adj(A) = adj(A) × A = det(A) × I
Frequently Asked Questions
What is the difference between minor and cofactor?
A minor Mᵢⱼ is just the determinant of the submatrix (no sign). A cofactor Cᵢⱼ = (-1)ⁱ⁺ʲ × Mᵢⱼ includes the sign factor based on position.
How do I know the sign for each cofactor?
The sign is (-1)ⁱ⁺ʲ. Starting from position (1,1) with +, it alternates in a checkerboard pattern: +, -, +, -, ...
What's the relationship between cofactor matrix and adjoint?
The adjoint (adjugate) is the transpose of the cofactor matrix: adj(A) = Cᵀ. It's used to find the inverse: A⁻¹ = adj(A) / det(A).
Can I use cofactor matrix to find the inverse?
Yes! First find the cofactor matrix C, then transpose it to get adj(A) = Cᵀ, then divide by determinant: A⁻¹ = adj(A) / det(A).
Why does the minor matrix have no signs?
The minor is just the determinant of the submatrix - it doesn't include the (-1)ⁱ⁺ʲ factor. The cofactor applies the sign pattern.