Adjugate matrix/Definition

Adjugate matrix

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

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

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