Minors and Cofactors

In matrix theory, a minor is the determinant of a matrix obtained by deleting a given row and column. A minor multiplied by ( (-1)^{i+j}) gives what is called a cofactor.

1.0What are the Minors and Cofactors of a Matrix?

Minor of an Element 

A minor of an element in a matrix is the determinant of the matrix, which is arrived at by deleting the row and column, on which the element lies. A small problem of order n × n for a given matrix as described above, if one may want to find the minors and cofactors of determinate elements, then:

• Delete the entire row and column of the element.
• Calculate the determinant of the resulting (n - 1)(n - 1) matrix. 
• The minor of an element is denoted by M_ij.
• M_ij = determinant of the matrix obtained after removing the i-th row and j-th column from the matrix.

Cofactor of an Element

The cofactor of an element is found by multiplying its minor with (−1)^i+j, where i is row no and j is column no of the element. The cofactor of the element a_ij is expressed by C_ij and given by:

$C_{ij}=(-1{)}^{i+j}{M}_{ij}$

Where M_ij is the minor part of a_ij.

Properties of Minors and Cofactors

• The cofactor of an element is connected to the minor by the factor (-1)^i+j.
• The determinant of a matrix can be calculated by expanding along any row or column using the cofactors.
• The determinant will be zero (0) if a row or column forms a linear combination of other rows or columns.

2.0Cofactor Expansion

The cofactor expansion, also known as the Laplace expansion, is a way of determining the determinant of a matrix. The determinant of a matrix A = [a_ij] can be expanded along any row or column. 

If we expand along the i-th row, the determinant is given by:

$det(A)={\sum }_{j=1}^n{a}_{ij}{c}_{ij}={\sum }_{j=1}^n{a}_{ij}(-1{)}^{i+j}{M}_{ij}$

Similarly, if we expand along the j-th column, the determinant changes to: 

$det(A)={\sum }_{i=1}^n{a}_{ij}{C}_{ij}={\sum }_{n=1}^n{a}_{ij}(-1{)}^{i+j}{M}_{ij}$

3.0Find Minors and Cofactors of the Elements of the Determinant

Problem 1: In the given matrix: 

       2  3  1

A = 4  5  7

       6  8  9

Find: 

• Find the minor M₁₁ of the element a₁₁ = 2. 
• Find the Cofactor C₁₁ of the element a₁₁ = 2.

Solution: 

1. Find the minor M₁₁ of the element a₁₁

To find Minor M₁₁ delete the first row and first column of the given matrix A, and we will get: 

$\left[\begin{array}{cc}5& 7\\ 8& 9\end{array}\right]$

$M_{11}=det\left[\begin{array}{cc}5& 7\\ 8& 9\end{array}\right]=(5\times 9)-(8\times 7)=45-56=-11$

2) Find the Cofactor C₁₁ of the element a₁₁

Using the formula for finding the cofactor; 

$C_{ij}=(-1{)}^{i+j}{M}_{ij}$

$C_{11}=(-1{)}^{1+1}\times (-11)=-11$

Problem 2: Given the matrix 

       1  0  2  –1

A = 3  1  1   2

       4  2  2   3

       0  1  1   4

Find: 

• Find the minor M_12​ for element a₁₂ = 0
• Find the cofactor C₁₂​ for element a₁₂ = 0

Solution: 

1. For finding M₁₂ delete the first row and second column: 

3  1  2

4  2  3

0  1  4

M12=det  3  1  2

                   4  2  3

                   0  1  4

$M_{12}=3\times det\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]-1\times det\left[\begin{array}{cc}4& 3\\ 0& 4\end{array}\right]+2\times det\left[\begin{array}{cc}4& 2\\ 0& 1\end{array}\right]$

$M_{12}=3(8-3)-1(16-0)+2(4-0)$

$M_{12}=15-16+8$

$M_{12}=7$

2. Find the cofactor C₁₂ for the element a₁₂ 

With the help of the formula: 

$C_{ij}=(-1{)}^{i+j}{M}_{ij}$

$C_{12}=(-1{)}^{1+2}\times 7$

$C_{12}=-7$

Problem 3: There is a company that has three departments such as marketing, sales, and production. The following matrix is used to represent the number of items sold by each department over three different years:

       3  5  2  

A = 4  7  3 

       6  8  4

Here:

$R_1$=Marketing department sales

$R_2$=Sales Department sales

$R_3$=Production department sales

Now, find the determinant of the above matrix A to see whether the given pattern of sales across the years is linearly dependent or independent. Sales data are linearly dependent if their determinant is zero, otherwise independent.

Solution: Calculate the determinant along R₁

$det(A)=3\times det\left[\begin{array}{cc}7& 3\\ 8& 4\end{array}\right]-5\times \left[\begin{array}{cc}4& 3\\ 6& 4\end{array}\right]+2\times \left[\begin{array}{cc}4& 7\\ 6& 8\end{array}\right]$

$det(A)=3(28-24)-5(16-18)+2(32-42)$

$det(A)=12+10-20=2$

Since the determinant of the matrix is not 0, so the sales data is not linearly dependent.