The cofactor matrix is the stepping stone to the adjugate and the inverse, and to computing determinants by expansion. For [[a, b], [c, d]] it is [[d, −c], [−b, a]].
How to use the Cofactor Matrix Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
For each position, the row and column are deleted, the determinant of what remains is computed and the checkerboard sign is applied.
Formula
Cᵢⱼ = (−1)^(i+j) · det(Mᵢⱼ)
Worked example
Inputs — Matrix size: 3 × 3; a₁: 4; b₁: 1; c₁: 2; a₂: 1; b₂: 3; c₂: 0; a₃: 2; b₃: 0; c₃: 5.
Results — Cofactor matrix = ⎡ 15 −5 −6 ⎤
⎢ −5 16 2 ⎥
⎣ −6 2 11 ⎦; Determinant = 43; Each cofactor = C₁₁ = (−1)² × 15 = 15
C₁₂ = (−1)³ × 5 = −5
C₁₃ = (−1)⁴ × −6 = −6
C₂₁ = (−1)³ × 5 = −5
C₂₂ = (−1)⁴ × 16 = 16
C₂₃ = (−1)⁵ × −2 = 2
C₃₁ = (−1)⁴ × −6 = −6
C₃₂ = (−1)⁵ × −2 = 2
C₃₃ = (−1)⁶ × 11 = 11.
Key terms
- Cofactor matrix
- The matrix of all cofactors.
Frequently asked questions
What is a cofactor?
A signed minor: (−1)^(i+j) times the determinant of the submatrix without row i and column j.
How do I get the inverse from it?
Transpose it (the adjugate) and divide by the determinant.
What is the cofactor matrix of a 1 × 1 matrix?
[1].
How do cofactors give the determinant?
Multiply each entry of one row by its cofactor and add.