Matrix Minor, Cofactor, and Adjugate Guide | Generated by AI
Here’s a clean way to untangle minor, cofactor, cofactor matrix, and adjugate so you never mix them again.
1. Minor
For an entry at position ((i,j)) in a matrix (A):
- Minor (M_{ij}) = determinant of the matrix you get after deleting row (i) and column (j).
- No signs involved.
- No (a_{ij}) involved.
So: [ M_{ij} = \det(A \text{ without row } i \text{ and column } j) ]
2. Cofactor
For the same entry:
- Cofactor (C_{ij}) = sign factor ( (-1)^{i+j} ) multiplied by the minor.
So: [ C_{ij} = (-1)^{i+j} M_{ij} ]
Still no multiplying by the original matrix entry (a_{ij}). That is only in the Laplace expansion of det(A), not in the definition of cofactors.
3. Cofactor Matrix
Make a matrix out of all the cofactors:
[
\text{Cof}(A) =
\begin{bmatrix}
C_{11} & C_{12} & \cdots
C_{21} & C_{22} & \cdots
\vdots & \vdots
\end{bmatrix}
]
Nothing is flipped here. Just list the cofactors in their original positions.
4. Adjugate (a.k.a. Adjoint)
The adjugate is simply the transpose of the cofactor matrix:
[ \operatorname{adj}(A) = \text{Cof}(A)^\mathsf{T} ]
That’s the only difference: you swap rows and columns.
5. Where does the (a_{ij}) multiplication appear?
You only multiply by (a_{ij}) when expanding determinant along a row or column:
[ \det(A) = \sum_{j=1}^n a_{ij} C_{ij} ]
or
[ \det(A) = \sum_{i=1}^n a_{ij} C_{ij} ]
That’s not part of the definition of the cofactor.
6. Quick mental summary (no tables)
- Remove row (i), column (j) → minor.
- Multiply by sign → cofactor.
- Collect all cofactors → cofactor matrix.
- Transpose that → adjugate.
- Multiply by (a_{ij}) only when computing determinant.
If you want, I can also give you a concrete 3×3 example that shows all four objects at once.