Definition:Cofactor/Element

From ProofWiki
Jump to navigation Jump to search

Definition

Let:

$D = \begin {vmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {vmatrix}$

be a determinant of order $n$.


Let $a_{r s}$ be an element of $D$.

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


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

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


Sources