Cofactor/Examples/Arbitrary Example 2

Example of Cofactor

Let $D$ be the determinant defined as:

$\quad D = \begin {vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end {vmatrix}$

Then the cofactor of $a_{2 1}$ is defined as:

$\ds A_{21}$

$=$

$\ds \paren {-1}^3 D_{21}$

$\ds $

$=$

$\ds \paren {-1}^3 \begin {vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end {vmatrix}$

$\ds $

$=$

$\ds -\paren {a_{12} a_{33} - a_{13} a_{32} }$

$\ds $

$=$

$\ds a_{13} a_{32} - a_{12} a_{33}$