Minors And Cofactors

A minor is the determinant of the small square matrix obtained by removing one row and column from the original matrix. Minors are used to find the determinant and cofactor of the matrix. For example, 

Solution:
In this matrix, we are going to remove the first row and column:
A =  85 96
Then find the determinant for the minor
det(A) = ad -bc
M11 = (5)(9) -(6)(8)
M11 = 45 -48
M11 =  -3
The minor of a11 is -3
 

The cofactor of an element is its minor multiplied by (-1)i+j, where i and j are the rows and columns of the elements. For example, find the cofactor of the element in row 1, column 1 (which is 1).

Solution:
In this matrix, we are going to remove the first row and column:
A =  85 96
Then find the determinant for the minor
det(A) = ad -bc
M11 = (5)(9) -(6)(8)
M11 = 45 -48
M11 =  -3
The minor of a11 is -3

Then find the cofactor
Cofactor = (-1)i+j × Mi+j
Cofactor = (-1)1+1 × (-3)
               = (+1) × (-3)
Cofactor = -3

The cofactor of 1 is -3
 

• The cofactor of an element obtained by the minor is connected to (-1)i+j.
• Cofactors are used in cofactor expansion to calculate the determinant along any row or column.
• If one row or column of the matrix is a linear combination of others, so the determinant of the matrix is zero
• The cofactors are used to form an adjoint matrix. It is important to find the inverse of a square matrix.
   

The cofactor expansion, also called Laplace expansion, is used to find the determinant of the matrix by using the minors and cofactors. It follows the formula: Det (A) = ai1Ci1 + ai2Ci2 +. . . + ain Cin
 

The adjoint of a matrix is the transpose of the cofactor matrix of the original matrix. There are several steps involved in finding an adjoint matrix. Find the cofactor of each element in the matrix. Create a new matrix by using all the cofactors, the new matrix is called the cofactor matrix. Then transpose the cofactor matrix, which gives the adjoint matrix.
 

The inverse of a matrix is the matrix that, when multiplied with the original matrix, gives the identity matrix, much like multiplying a number by its reciprocal.
 

Minors and cofactors are not only used in solving problems, but they also play a role in the real world. Here are some examples are given below

• Engineering: Engineers use the minors and cofactors to check that the building or machine is strong and balanced. This helps to make sure things won’t break or fall.
• Computer Graphics and Animation: The animators and graphic designers use the matrix to rotate and transform 3D objects in games or movies.
• Architecture: Designing buildings or bridges, architects use the matrix to calculate the load.
• Robotics: In hospitals, there is a need for robots to move their arms for surgery. Engineers use the matrix to control the movement.
• Navigation and GPS Systems: To find the best path or location, GPS systems solve complex math using matrices.