Cofactor Matrix — Definition, Formula & Examples
Cofactor Matrix
Matrix of Cofactors
A matrix with elements that are the cofactors, term-by-term, of a given square matrix.
![Example finding cofactor matrix of A=[1,2,3;0,4,5;1,0,6], showing cofactor calculations A11=24 through A33=4, resulting in...](c_assets/c46a.gif)
See also
Key Formula
Cij=(−1)i+jMij
Where:
- Cij = The cofactor of the element in row i, column j
- (−1)i+j = The sign factor, which alternates in a checkerboard pattern
- Mij = The minor — the determinant of the submatrix formed by deleting row i and column j
Worked Example
Problem: Find the cofactor matrix of A = [[2, 3, 1], [0, 4, 5], [1, 0, 6]].
Step 1: Recall the sign pattern for a 3×3 matrix. The sign of cofactor C_ij is (+) when i+j is even and (−) when i+j is odd.
Signs=+−+−+−+−+
Step 2: Compute cofactor C₁₁: delete row 1 and column 1, find the determinant, and apply the sign (+).
C11=+4056=+(24−0)=24
Step 3: Compute C₁₂: delete row 1 and column 2, apply the sign (−).
C12=−0156=−(0−5)=5
Step 4: Compute C₁₃: delete row 1 and column 3, apply the sign (+).
C13=+0140=+(0−4)=−4
Step 5: Compute the second row of cofactors: C₂₁, C₂₂, C₂₃.
C21=−3016=−(18−0)=−18,C22=+2116=11,C23=−2130=−(0−3)=3
Step 6: Compute the third row of cofactors: C₃₁, C₃₂, C₃₃.
C31=+3415=11,C32=−2015=−10,C33=+2034=8
Step 7: Assemble all cofactors into the cofactor matrix C.
C=24−1811511−10−438
Answer: The cofactor matrix is C = [[24, 5, −4], [−18, 11, 3], [11, −10, 8]].
Frequently Asked Questions
What is the difference between a cofactor matrix and the adjugate (classical adjoint)?
The adjugate (or classical adjoint) is the transpose of the cofactor matrix. If C is the cofactor matrix, then adj(A) = Cᵀ. You need the adjugate — not the cofactor matrix directly — when computing the inverse of a matrix using the formula A⁻¹ = adj(A)/det(A).
How do you find the cofactor matrix of a 2×2 matrix?
For a 2×2 matrix [[a, b], [c, d]], each minor is just a single element. Applying the checkerboard signs gives the cofactor matrix [[d, −c], [−b, a]]. Notice this is closely related to the well-known 2×2 inverse formula.
Cofactor Matrix vs. Adjugate (Classical Adjoint)
The cofactor matrix places each cofactor C_ij in position (i, j). The adjugate takes that matrix and transposes it, placing C_ij in position (j, i). The adjugate is the version used directly in the matrix inverse formula A⁻¹ = adj(A)/det(A).
Why It Matters
The cofactor matrix is essential for computing the inverse of a matrix by the adjugate method, which is especially useful in theoretical proofs and symbolic computation. It also underlies cofactor expansion, the standard technique for evaluating determinants of matrices larger than 2×2. In Cramer's Rule, the cofactors appear directly in the formulas for solving systems of linear equations.
Common Mistakes
Mistake: Forgetting the sign factor (−1)^(i+j) and using the minor directly as the cofactor.
Correction: Always apply the checkerboard sign pattern. A positive minor can become a negative cofactor if i+j is odd, and vice versa.
Mistake: Using the cofactor matrix where the adjugate is needed (e.g., in the inverse formula).
Correction: The inverse formula requires the transpose of the cofactor matrix (the adjugate). Remember: adj(A) = Cᵀ, then A⁻¹ = Cᵀ / det(A).
Related Terms
- Cofactor — Individual entry of the cofactor matrix
- Matrix — General structure the cofactor matrix is built from
- Square Matrix — Only square matrices have cofactor matrices
- Classical Adjoint — Transpose of the cofactor matrix
- Inverse of a Matrix — Computed using the adjugate from cofactors
- Element of a Matrix — Each element generates one cofactor
- Determinant — Minors used in cofactors are determinants