Adjugate Matrix

The adjugate matrix, also known as the classical adjugate or adjunct matrix, is a matrix that plays a crucial role in various areas of mathematics, physics, and engineering.

Definition

Given a square matrix $A$ of order $n\times n$, the adjugate matrix of $A$, denoted by $\mathrm{adj}(A)$, $\mathrm{Adj}(A)$, $A^{\ast }$ or $A^{\vee }$ is an $n\times n$ matrix whose elements are the cofactors of the elements of $A$.

Cofactors: The cofactor of an element $a_{ij}$ of a matrix $A$ is defined as:

$$C_{ij}=(-1{)}^{(i+j)}\times M_{ij}$$

where $M_{ij}$ is the minor of $a_{ij}$, which is the determinant of the $(n-1)\times (n-1)$ submatrix obtained by removing the $i$-th row and $j$-th column of $A$.

And we define $\mathrm{adj}(A)$ as the transpose of $C$, that is, $\mathrm{adj}(A)=C^T$

Properties

1. Inverse: The adjugate matrix is related to the inverse of $A$ by the formula: $A^{-1}=\frac{1}{\det (A)}\times \mathrm{adj}(A)$