How this calculator works
The minor of an entry deletes its row and column and takes the determinant of what remains. The cofactor multiplies that minor by plus one or minus one using the checkerboard pattern: plus in the top left corner, then alternating along rows and columns. The cofactor matrix collects all cofactors in the original positions. Its transpose is the adjugate, which satisfies A times adj(A) equals det(A) times the identity.
Formula
Cij = (-1)^(i+j) det(Mij), where Mij deletes row i and column j. adj(A) = C transposed.
Worked example
For the matrix with rows 1, 2 and 3, 4, the cofactors are 4, -3, -2 and 1. The cofactor matrix is 4, -3; -2, 1, and its transpose is the adjugate 4, -2; -3, 1. The determinant is 1x4 - 2x3 = -2.
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Frequently asked questions
Is the cofactor matrix the same as the adjugate?
No. The adjugate is the transpose of the cofactor matrix. For a square matrix, A times its adjugate equals the determinant times the identity.
Can a cofactor be zero?
Yes. A cofactor is zero whenever the minor determinant is zero, which often happens when rows or columns are dependent.
Where does the sign pattern come from?
The sign is the parity of the row plus column index. The pattern starts with plus at the top left and alternates like a checkerboard.