Mathwords logoMathwords

Cofactor Matrix: Definition, Sign Pattern & How to Build It

Cofactor Matrix
Matrix of Cofactors

A matrix with elements that are the cofactors, term-by-term, of a given square matrix.

See also

Adjoint, inverse of a matrix

Key Formula

Cij=(−1)i+j MijC_{ij} = (-1)^{i+j} \, M_{ij}
Where:
  • CijC_{ij} = The cofactor of the element in row i, column j
  • (−1)i+j(-1)^{i+j} = The sign factor, which alternates in a checkerboard pattern
  • MijM_{ij} = 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=[+−+−+−+−+]\text{Signs} = \begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}
Step 2: Compute cofactor C₁₁: delete row 1 and column 1, find the determinant, and apply the sign (+).
C11=+∣4506∣=+(24−0)=24C_{11} = +\begin{vmatrix} 4 & 5 \\ 0 & 6 \end{vmatrix} = +(24 - 0) = 24
Step 3: Compute C₁₂: delete row 1 and column 2, apply the sign (−).
C12=−∣0516∣=−(0−5)=5C_{12} = -\begin{vmatrix} 0 & 5 \\ 1 & 6 \end{vmatrix} = -(0 - 5) = 5
Step 4: Compute C₁₃: delete row 1 and column 3, apply the sign (+).
C13=+∣0410∣=+(0−4)=−4C_{13} = +\begin{vmatrix} 0 & 4 \\ 1 & 0 \end{vmatrix} = +(0 - 4) = -4
Step 5: Compute the second row of cofactors: C₂₁, C₂₂, C₂₃.
C21=−∣3106∣=−(18−0)=−18,C22=+∣2116∣=11,C23=−∣2310∣=−(0−3)=3C_{21} = -\begin{vmatrix} 3 & 1 \\ 0 & 6 \end{vmatrix} = -(18-0) = -18, \quad C_{22} = +\begin{vmatrix} 2 & 1 \\ 1 & 6 \end{vmatrix} = 11, \quad C_{23} = -\begin{vmatrix} 2 & 3 \\ 1 & 0 \end{vmatrix} = -(0-3) = 3
Step 6: Compute the third row of cofactors: C₃₁, C₃₂, C₃₃.
C31=+∣3145∣=11,C32=−∣2105∣=−10,C33=+∣2304∣=8C_{31} = +\begin{vmatrix} 3 & 1 \\ 4 & 5 \end{vmatrix} = 11, \quad C_{32} = -\begin{vmatrix} 2 & 1 \\ 0 & 5 \end{vmatrix} = -10, \quad C_{33} = +\begin{vmatrix} 2 & 3 \\ 0 & 4 \end{vmatrix} = 8
Step 7: Assemble all cofactors into the cofactor matrix C.
C=[245−4−1811311−108]C = \begin{bmatrix} 24 & 5 & -4 \\ -18 & 11 & 3 \\ 11 & -10 & 8 \end{bmatrix}
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