Cofactors

The cofactor expansion computes determinants recursively:

$$\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} C_{ij}$$

Where $C_{ij}$ is the cofactor — the determinant of the $(n-1) \times (n-1)$ submatrix obtained by deleting row $i$ and column $j$.

Sign Pattern

$$\begin{pmatrix} + & - & + & \cdots \\ - & + & - & \cdots \\ + & - & + & \cdots \end{pmatrix}$$

Cofactor Matrix

The cofactor matrix $C$ has entries $C_{ij}$. Then:

$$A^{-1} = \frac{1}{\det(A)} C^T$$

For Inverse

This gives the adjugate formula:

$$A^{-1} = \frac{\text{adj}(A)}{\det(A)}$$

Where $\text{adj}(A) = C^T$ is the adjugate matrix.