Minors and Cofactors

Minors and cofactors are two of the most important concepts in matrices. They are essential in finding the adjoint and the inverse of a matrix.

Minors and cofactors are defined for each element of the matrix.

Minors

The minor of an element of the matrix is equal to the determinant of the remaining elements of the matrix, obtained after deleting the row and column containing the particular element in the matrix.

Minor of an element $a_{ij}$ of a determinant is the determinant obtained by deleting its $i$th row and $j$th column in which element $a_{ij}$ occurs. Minor of an element $a_{ij}$ is denoted by $M_{ij}$

Example

Find the minor of element $4$ in the determinant ${\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}}$

The element $4$ occurs in $2$^nd row and $1$^st column.

Delete the elements in $2$^nd row and $1$^st column , the remaining elements will be the minor of $4$ represented by $M_{21}$

$M_{21}={\begin{vmatrix}5&6\\8&9\end{vmatrix}}=5\times 9-6\times 8=45-48=-3$

Cofactors

The cofactor of an element of the matrix is obtained by multiplying the minor of the element with $-1$ to power of the sum of $i$ ^th and $j$ ^th column containing the element.

Cofactor of an element $a_{ij}$, denoted by $A_{ij}$ is defined by $A_{ij}$ = $(-1)^{i+j}$$M_{ij}$ , where $M_{ij}$is minor of $a_{ij}$

Example

Find the cofactor of element $4$ in the determinant ${\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}}$

The element $4$ occurs in $2$^nd row and $1$^st column. Hence $a_{21}=4$

Cofactor of an element $a_{21}$, denoted by $A_{21}$ is defined by $A_{21}=(-1)^{2+1}M_{21}$ , where $M_{21}$ is minor of $a_{21}$

We know from the above $M_{21}=-3$

$A_{21}=(-1)^{2+1}\times -3=(-1)^{3}\times -3=-1\times -3=3$