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C

Cofactor Matrix

Cofactor Cᵢⱼ = (−1)^(i+j) det(Mᵢⱼ), where Mᵢⱼ is the minor (submatrix with row i, col j removed). The adjugate is the transpose of the cofactor matrix; A⁻¹ = adj(A)/det(A).

Concept Fundamentals
Cᵢⱼ=(−1)^(i+j)det(Mᵢⱼ)
Formula
Mᵢⱼ = submatrix
Minor
adj(A)=(cof(A))ᵀ
Adjugate
adj(A)/det(A)
A⁻¹
Compute Cofactor MatrixCᵢⱼ = (−1)^(i+j) det(Mᵢⱼ)

Why This Mathematical Concept Matters

Why: Cofactors build the adjugate for matrix inverse. Used in Cramer's rule and determinant expansion.

How: For each (i,j), remove row i and col j to get Mᵢⱼ. Cᵢⱼ = (−1)^(i+j) det(Mᵢⱼ). Sign pattern alternates like a checkerboard.

  • ●adj(A) = (cof(A))ᵀ.
  • ●A·adj(A) = det(A)·I.
  • ●Sign: + − + − … (checkerboard).
Sources:Gilbert Strang, Linear AlgebraWolfram MathWorld

Quick Examples — Click to Load

Matrix A (3×3)

cofactor_matrix.sh
CALCULATED
$ cofactor --size="3×3"
Size
3×3
Sign pattern
+ − + | − + − | + − +
Original A (3×3)
000
000
000
Cofactor cof(A)
000
000
000
Share:
Cofactor Matrix
3×3 cof(A)
[0, 0, 0] | [0, 0, 0] | [0, 0, 0]
numbervibe.com/calculators/mathematics/linear-algebra/cofactor-matrix-calculator

Cofactor Element Distribution (Bar)

Sign Distribution (Doughnut)

Calculation Steps

SETUP
Original A[0, 0, 0] [0, 0, 0] [0, 0, 0]
SETUP
Size3×3
FORMULA
FormulaC_ij = (-1)^(i+j) × det(M_ij)
FORMULA
M_ij = submatrix (remove row i, col j)Minor determinant
COFACTORS
C(1,1)(-1)^2×det(M)=0
COFACTORS
C(1,2)(-1)^3×det(M)=0
COFACTORS
C(1,3)(-1)^4×det(M)=0
COFACTORS
C(2,1)(-1)^3×det(M)=0
COFACTORS
C(2,2)(-1)^4×det(M)=0
COFACTORS
C(2,3)(-1)^5×det(M)=0
COFACTORS
C(3,1)(-1)^4×det(M)=0
COFACTORS
C(3,2)(-1)^5×det(M)=0
COFACTORS
C(3,3)(-1)^6×det(M)=0
RESULT
Cofactor matrix cof(A)[0, 0, 0] [0, 0, 0] [0, 0, 0]

⚠️For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

det(A) = Σⱼ a₁ⱼ C₁ⱼ (expand row 1)

⚡

adj(AB) = adj(B) adj(A)

Key Takeaways

  • • Cofactor Cij = (-1)i+j × det(Mij), Mij = minor (submatrix without row i, col j).
  • • Cofactor matrix: replace each element by its cofactor.
  • • adj(A) = (cof(A))ᵀ — adjugate is transpose of cofactor matrix.
  • • Laplace expansion: det(A) = Σ aij Cij along any row or column.
  • • Sign pattern: checkerboard + − + − from (1,1).

Did You Know?

📐2×2: cof([[a,b],[c,d]]) = [[d,-c],[-b,a]].Source: Shortcut
🧮Laplace expansion uses cofactors for determinant.Source: Laplace
⚛️Cofactors appear in inverse: A⁻¹ = adj(A)/det(A).Source: Inverse
📊Zero matrix: all cofactors are 0.Source: Special
🔢Identity: cof(I) = I for any size.Source: Identity
🎯Cramer's rule uses cofactors for Ax = b.Source: Cramer

How It Works

For each element (i,j): remove row i and column j to get minor Mij, compute det(Mij), multiply by (-1)i+j. Place result at (i,j) in cofactor matrix.

C_ij = (-1)^(i+j) × det(M_ij)

adj(A) = (cof(A))ᵀ

Expert Tips

2×2 Shortcut

[[a,b],[c,d]] → cof = [[d,-c],[-b,a]].

Sign Pattern

Checkerboard: + − + − from top-left.

Row/Column Expansion

Pick row/col with most zeros for fewer ops.

To Adjoint

Transpose cofactor matrix → adjugate.

Comparison Table

FeatureThis CalculatorNumPyManual
Minor + Cofactor steps✅❌⚠️
Bar & Doughnut charts✅❌❌
8 preset examples✅❌❌
Step-by-step✅❌⚠️

FAQ

What is the cofactor?

C_ij = (-1)^(i+j) × det(M_ij), M_ij = minor (submatrix).

What is the minor?

Determinant of submatrix with row i and column j removed.

Sign pattern?

Checkerboard: + at (1,1), − at (1,2), etc.

Relation to adjugate?

adj(A) = transpose of cofactor matrix.

Relation to inverse?

A⁻¹ = adj(A)/det(A) = (cof(A))ᵀ/det(A).

Zero matrix cofactor?

All cofactors are 0.

Identity matrix cofactor?

cof(I) = I for any size.

Laplace expansion?

det(A) = Σ a_ij C_ij along any row i (or column j).

Stats

C=(-1)^(i+j)M
Formula
n² cofactors
Elements
O(n⁴)
Complexity
adj=(cof)ᵀ
Adjoint

Sources

Disclaimer: For educational purposes. Uses JavaScript floating-point. Verify critical calculations independently.

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