Cofactor Matrix Calculator
Compute the cofactor matrix of a square matrix. Each entry C[i][j] is the signed minor — the determinant of the submatrix formed by deleting row i and column j, multiplied by (−1)^(i+j).
Notes
What is the Cofactor Matrix?
The minor M[i][j] is the determinant of the submatrix obtained by deleting row i and column j. The cofactor C[i][j] is the signed minor.
The sign pattern for a 3×3 cofactor matrix:
| + | − | + |
| − | + | − |
| + | − | + |
Example: C₁₁ and C₁₂ for a 3×3 Matrix
For A = [[1,2,3],[4,5,6],[7,8,9]], find C₁₁:
Delete row 1 and col 1, compute det of remaining 2×2:
| 5 | 6 |
| 8 | 9 |
M₁₁ = 5·9 − 6·8 = 45 − 48 = −3. Sign = (+1)^(1+1) = +1. So C₁₁ = +1 · (−3) = −3.
- Matrix Calculations Guide — Cofactor Matrix & Adjugate — In-depth notes on Cofactor Matrix & Adjugate with worked examples
Frequently Asked Questions
What is the minor M[i][j]?
The minor M[i][j] is the determinant of the submatrix formed by deleting row i and column j from A.
How does the cofactor relate to the inverse?
The inverse of A is adj(A)/det(A), where adj(A) is the transpose of the cofactor matrix.
How is the cofactor used in Laplace expansion?
det(A) = Σⱼ A[1][j] · C[1][j] — expanding along row 1. You can expand along any row or column using the corresponding cofactors.