# Minors and Cofactors

In the determinant

Î” = ...(1)

if we leave the row and the column passing through the element

*a*then the second-order determinant thus obtained is called the minor of_{ij}*a*and it is denoted by_{ij}*M*. Thus, we can get nine minors corresponding to the nine elements._{ij}For example, in determinant (1), the minor of the element

*a*

_{21}=

In terms of the notation of minors if we expand the determinant along the first row, then

Î” = (â€“1)

^{1+1}*a*_{11}*M*_{11}+ (â€“1)^{1+2}*a*_{12}*M*_{12}+ (â€“1)^{1+3}*a*_{13}*M*_{13}=

*a*_{11}*M*_{11}â€“*a*_{12}*M*_{12}+*a*_{13}*M*_{13}Similarly, expanding Î” along the second column, we have

Î” = â€“

*a*_{12}*M*_{12}+*a*_{22}*M*_{22}â€“*a*_{32}*M*_{32}The minor

*M*multiplied by (â€“1)_{ij}^{i}^{+j}is called the cofactor of the element*a*. If we denote the cofactor of the element_{ij}*a*, by_{ij}*C*, then the cofactor of_{ij}*a*is_{ij}*C*= (â€“1)_{ij}^{i}^{+j}*M*._{ij}Cofactor of the element

*a*_{21}is*C*

_{21}= (â€“1)

^{2+1}

*M*

_{21}=

In terms of the notation of the cofactors, we have

Î” =

*a*_{11}*C*_{11}+*a*_{12}*C*_{12}+*a*_{13}*C*_{13}=

*a*_{21}*C*_{21}+*a*_{22}*C*_{22}+*a*_{23}*C*_{23}=

*a*_{31}*C*_{31}+*a*_{32}*C*_{32}+*a*_{33}*C*_{33}Also,

*a*_{11}*C*_{21}+*a*_{12}*C*_{22}+*a*_{13}*C*_{23}= 0,*a*_{11}*C*_{31}+*a*_{12}*C*_{32}+*a*_{13}*C*_{33}= 0, etc.