Adjugate matrix

Definition

Let $K$ be a commutative ring, let $n \ge 2$, and let $A$ be an $n \times n$ matrix over $K$ with cofactors $C_{ij}$. The adjugate of $A$ is the $n \times n$ matrix $\operatorname{adj}(A)$ with entries $$\operatorname{adj}(A)_{ij} = C_{ji}.$$ So $\operatorname{adj}(A)$ is the transpose of the matrix of cofactors $(C_{ij})$.

Remarks

For $n = 2$, $\operatorname{adj} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$. The adjugate satisfies $A \operatorname{adj}(A) = \det(A) I_n$. Older texts call it the classical adjoint; it is not the adjoint of an inner product space.

Explanations

L1 human

To build the adjugate, replace each entry by its cofactor, and then transpose. The adjugate is almost the inverse: it is the inverse multiplied by the determinant. For $2 \times 2$ matrices, swap the diagonal entries and negate the other two.