# Definition:Cofactor/Element

## Definition

Let:

$D = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn}\end{vmatrix}$

be a determinant of order $n$.

Let $a_{rs}$ be an element of $D$.

Let $D_{rs}$ be the determinant of order $n-1$ obtained from $D$ by deleting row $r$ and column $s$.

Then the cofactor $A_{rs}$ of the element $a_{rs}$ is defined as:

$A_{rs} := \paren {-1}^{r + s} D_{rs}$