Compute the cofactor matrix and adjoint (adjugate) of any square matrix — with every minor determinant expanded step by step.
Matrix Setup
Results at a Glance
Step 1 — Sign Pattern
Step 2 — Cofactor Matrix C
Each entry $C_{ij} = (-1)^{i+j} \cdot M_{ij}$ where $M_{ij}$ is the determinant of the submatrix formed by deleting row $i$ and column $j$.
Click any cell in the cofactor matrix below to see its detailed calculation.
Detailed Calculations for Each Cofactor
Step 3 — Adjoint Matrix = Cᵀ
The adjoint (adjugate) of $A$ is the transpose of the cofactor matrix:
$$\text{adj}(A) = C^T \quad \Longrightarrow \quad [\text{adj}(A)]_{ij} = C_{ji}$$
Bonus — Inverse via Adjoint
Theory & Key Formulas
Minor $M_{ij}$ — The determinant of the $(n{-}1)\times(n{-}1)$ submatrix obtained by deleting row $i$ and column $j$ from $A$.
Cofactor $C_{ij}$ — The signed minor:
$$C_{ij} = (-1)^{i+j} M_{ij}$$
The sign pattern follows the checkerboard: $+$ when $i+j$ is even, $-$ when $i+j$ is odd.
Adjoint (Adjugate) — Transpose of the cofactor matrix:
$$\text{adj}(A) = C^T, \quad [\text{adj}(A)]_{ij} = C_{ji}$$