Minors And Cofactors

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

For example, \(\begin{bmatrix} 1&2&3\\4&5&6\\7&8&9 \end{bmatrix}\)

Solution: In this matrix, we are going to remove the first row and column:

\(M_{11} = \begin{bmatrix} 5&6\\8&9 \end{bmatrix}\)

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 the multiplication of its minor 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).

​​\(\begin{bmatrix} 1&2&3\\4&5&6\\7&8&9 \end{bmatrix}\)

Solution:
 

1. In this matrix, we are going to remove the first row and column:
   \(\begin{bmatrix} 5&6\\8&9 \end{bmatrix}\)
    
2. 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
    
3. Then find the cofactor
   Cofactor = (-1)^i+j × M_i+j
   Cofactor = (-1)¹⁺¹ × (-3)
                  = (+1) × (-3)
   Cofactor = -3

The cofactor of 1 is -3
 

The matrix consisting of all the cofactors of the element is called the cofactor matrix.

Now, let's see some properties of minor and cofactors. The properties are:

1. The cofactor of an element obtained by the minor is connected to (-1)^i+j.
    
2. Cofactors are used in cofactor expansion to calculate the determinant along any row or column.
    
3. If one row or column of the matrix is a linear combination of others, so the determinant of the matrix is zero.
    
4. 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: \(Det (A) = a_{i1}Ci_{i1} + a_{i2}C_{i2} +. . . + a_{in} C_{in}\)
 

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. The steps are as follows:

1. Find the cofactor of each element in the matrix.
    
2. Create a new matrix by using all the cofactors, the new matrix is called the cofactor matrix.
    
3. 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. The formula \(A^{-1} = \frac{1}{|A|} Adj A\) can be used to find the inverse of a matrix.

To help to master minors and cofactors, here are some essential tips and tricks:

1. To find the determinant of a 2 × 2 matrix, multiply the diagonal elements and subtract the product of non-diagonal elements.
    
2. The minor and determinant of 1 × 1 matrix is the same and is equal to the element itself.
    
3. Use the sign pattern, to remember the correct sign for finding cofactors.
    
4. If all the elements in a row or column are zero, then its determinant is zero.
    
5. The inverse of an identity matrix is itself.

Parent Tip: 

• Encourage your child to practice minors and cofactors with matrix of smaller order, then mover to larger.
• You can do quizes to help your child remember all the formulas.

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

1. 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.
    
2. Computer Graphics and Animation: The animators and graphic designers use the matrix to rotate and transform 3D objects in games or movies.
    
3. Architecture: Designing buildings or bridges, architects use the matrix to calculate the load.
    
4. Robotics: In hospitals, there is a need for robots to move their arms for surgery. Engineers use the matrix to control the movement.
    
5. Navigation and GPS Systems: To find the best path or location, GPS systems solve complex math using matrices.