Adjugate matrix/Definition

For a square matrix ${\displaystyle {}M\in \operatorname {Mat} _{n}(K)}$, we call

${\displaystyle M^{\operatorname {adj} }=(b_{ij}){\text{ with }}b_{ij}=(-1)^{i+j}\det M_{ji}}$

the adjugate matrix of ${\displaystyle {}M}$, where ${\displaystyle {}M_{ji}}$ arises from ${\displaystyle {}M}$ by deleting the ${\displaystyle {}j}$-th row and the ${\displaystyle {}i}$-th column.