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Cofactors & Adjugate Matrix

Adjugate matrix calculator. Enter an n×n matrix to compute its cofactors and build adj(A), showing intermediate cofactors—for linear algebra practice.

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adj(A) = (cofactor)ᵀ, A·adj(A) = det(A)·I
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Cofactors & Adjugate Matrix

1. Definition
The transpose of the cofactor matrix is the adjugate adj(A), satisfying A·adj(A)=det(A)·I, and A⁻¹=adj(A)/det(A).
2. Symbols
SymbolMeaning
Cᵢⱼcofactor = (−1)^(i+j)·minor
adj(A)transpose of cofactor matrix
det(A)determinant
3. How this tool works
  • Minor Mᵢⱼ = determinant removing row i, column j;
  • Cofactor Cᵢⱼ = (−1)^(i+j)·Mᵢⱼ;
  • adj(A) = (Cᵢⱼ)ᵀ.
4. Steps
  1. Enter A;
  2. Click Calculate for adj(A) and the verification.
5. Example
Example: A=1 2;3 4. C=4 −3;−2 1, adj(A)=4 −2;−3 1, and A·adj(A)=−2·I.
6. Pitfalls
The adjugate is the transpose of the cofactor matrix;
Watch the (−1)^(i+j) sign;
adj(A) exists even when det A=0.

❓ FAQ｜Cofactors & Adjugate Matrix

How is the adjugate defined?
The transpose of the cofactor matrix.
Relation to the inverse?
A⁻¹ = adj(A)/det(A).
Minor vs cofactor?
The cofactor includes a (−1)^(i+j) sign.
2×2 formula?
For [[a,b],[c,d]], adjugate = [[d,−b],[−c,a]].
Does adj(A) exist when det A=0?
Yes, but A is not invertible.