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 Mij.
- Mij = 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 aij is expressed by Cij and given by:
Cij=(−1)i+jMij
Where Mij is the minor part of aij.
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 = [aij] can be expanded along any row or column.
If we expand along the i-th row, the determinant is given by:
det(A)=∑j=1naijcij=∑j=1naij(−1)i+jMij
Similarly, if we expand along the j-th column, the determinant changes to:
det(A)=∑i=1naijCij=∑n=1naij(−1)i+jMij
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:
- 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:
[5879]
M11=det[5879]=(5×9)−(8×7)=45−56=−11
2) Find the Cofactor C11 of the element a11
Using the formula for finding the cofactor;
Cij=(−1)i+jMij
C11=(−1)1+1×(−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 M12 for element a12 = 0
- Find the cofactor C12 for element a12 = 0
Solution:
- 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
M12=3×det[2134]−1×det[4034]+2×det[4021]
M12=3(8−3)−1(16−0)+2(4−0)
M12=15−16+8
M12=7
- Find the cofactor C12 for the element a12
With the help of the formula:
Cij=(−1)i+jMij
C12=(−1)1+2×7
C12=−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:
R1=Marketing department sales
R2=Sales Department sales
R3=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×det[7834]−5×[4634]+2×[4678]
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.