Adjugate Matrix

linear-algebra

Definition

Adjugate Matrix

Let $A=(a_{ij})\in K^{n\times n}$ be a square matrix over a field $K$. The adjugate matrix of $A$, denoted by $\mathrm{adj}(A)$, is the transpose of the cofactor matrix of $A$. Thus its $(i,j)$-entry is given by

$$(\mathrm{adj}(A){)}_{ij}=(-1{)}^{i+j}\det (A_{ji}),$$

where $A_{ji}$ is the submatrix obtained from $A$ by deleting the $j$-th row and the $i$-th column.

It satisfies the identity

$$A\mathrm{adj}(A)=\mathrm{adj}(A)A=\det (A)I.$$

In particular, if $\det (A)\ne 0$, then $A$ is invertible and

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

Example: For

$$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right),$$

one has

$$\mathrm{adj}(A)=\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right).$$