Definition of adjugate/classical adjoint of a matrix

Definition: The adjugate (or classical adjoint) of a matrix $A$, denoted $\text{adj}(A)$, is the transpose of the cofactor matrix: $\text{adj}(A)_{ij} = C_{ji}$. The key property is $A \cdot \text{adj}(A) = \det(A) \cdot I$, which gives the formula $A^{-1} = \frac{1}{\det(A)}\text{adj}(A)$ for invertible matrices.