Cofactor and Minor: Definitions

Cofactor

In linear algebra, the cofactor (sometimes called adjunct) describes a particular construction that is useful for calculating both the determinant and inverse of square matrices. Specifically the cofactor of the $(i,j)$ entry of a matrix, also known as the $(i,j)$ cofactor of that matrix, is the signed minor of that entry.

The cofactor of $a_{ij}$ entry of a matrix is defined as:

$\displaystyle C_{ij}=(-1)^{i+j}M_{ij}$

Minor

To know what the signed minor is, we need to know what the minor of a matrix is. In linear algebra, a minor of a matrix $A$ is the determinant of some smaller square matrix, cut down from $A$ by removing one or more of its rows or columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors.

Let $A$ be an $m \times n$ matrix and $k$ an integer with $0<k\leq m$, and $k \leq n$.  A $k \times k$ minor of $A$ is the determinant of a $k \times k$ matrix obtained from $A$ by deleting $m-k$ rows and $n-k$ columns.

Calculate the Determinant

The determinant of any matrix can be found using its signed minors. The determinant is the sum of the signed minors of any row or column of the matrix scaled by the elements in that row or column.

Calculating the Minors

The following steps are used to find the determinant of a given minor of a matrix A:

1. Choose an entry $a_{ij}$ from the matrix.
2. Cross out the entries that lie in the corresponding row $i$ and column $j$.
3. Rewrite the matrix without the marked entries.
4. Obtain the determinant of this new matrix.

$M_{ij}$ is termed the minor for entry $a_{ij}$.

Note: If $i+j$ is an even number, the cofactor coincides with its minor: $C_{ij}=M_{ij}$. Otherwise, it is equal to the additive inverse of its minor: $C_{ij}=-M_{ij}$

Calculating the Determinant

We will find the determinant of the following matrix A by calculating the determinants of its cofactors for the third, rightmost column and then multiplying them by the elements of that column.

 $\displaystyle \begin{bmatrix} 1 & 4 & 7\\ 3 & 0 & 5\\ -1& 9 &11\\ \end{bmatrix}$

As an example, we will calculate the determinant of the minor $M_{23}$, which is the determinant of the $2 \times 2$ matrix formed by removing the $2$nd row and $3$rd column.  A black dot represents an element we are removing. 

$\displaystyle \begin{aligned} \begin{vmatrix} 1 & 4 & \bullet\\ \bullet& \bullet& \bullet\\ -1& 9 &\bullet \end{vmatrix} &= \begin{vmatrix} 1 & 4\\ -1&9 \end{vmatrix}\\ &=(9-(-4))\\&=13 \end{aligned}$

Since $i+j=5 $ is an odd number, the cofactor is the additive inverse of its minor: $-(13)=-13$

We multiply this number by $a_{23}=5$, giving $-65$.

The same process is carried out to find the determinants of $C_{13}$ and $C_{33}$, which are then multiplied by $a_{13}$ and $a_{33}$, respectively. The determinant is then found by summing all of these:

$\begin {aligned} \det{A} &= a_{13}\det{C_{13}}+a_{23}\det{C_{23}}+a_{33}\det{C_{33}} \\ &= 7\cdot27-5\cdot13+11\cdot-12 \\&=-8 \end{aligned}$