Cofactor/Examples/Arbitrary Example 4

Example of Cofactor

Let $D$ be the determinant defined as:

$D = \begin {vmatrix} a & b & c \\ d & e & f \\ g & h & i \end {vmatrix}$

The cofactor of $e$ is defined as:

$\ds A_e$

$=$

$\ds A_{22}$

$\ds $

$=$

$\ds \paren {-1}^4 D_{22}$

$\ds $

$=$

$\ds \begin {vmatrix} a & c \\ g & i \end {vmatrix}$

$\ds $

$=$

$\ds \paren {a i - g c}$

The cofactor of $d$ is defined as:

$\ds A_d$

$=$

$\ds A_{12}$

$\ds $

$=$

$\ds \paren {-1}^3 D_{12}$

$\ds $

$=$

$\ds -\begin {vmatrix} b & c \\ h & i \end {vmatrix}$

$\ds $

$=$

$\ds \paren {c h - b i}$