Cofactor Matrix

A matrix formed using the cofactors of its elements is called a cofactor matrix.

• To find the cofactor of an element, we need to multiply its minor \((M^{ij}) \ by \ (-1)^{(i + j)}\).

• Here, i and j are the positional values, that is, the row number and column number of that element. 

• If the minor of an element is M_ij, its cofactor ( C_ij ) is:
  
  \(C^{ij} = (-1)^{(i + j)} × M^{ij}\)

To obtain the cofactor matrix, we first find the minor of each element in the original matrix. Then, we use those minors to calculate the cofactors.

For the minor matrix (M), calculate the minor for each element of the original matrix -

\(M = \begin{bmatrix} M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} &M_{23} \\ M_{31} & M_{32} &M_{33} \end{bmatrix}\)

​​\(\ Cofactor \ Matrix = \begin{bmatrix} +M_{11} & -M_{12} & +M_{13} \\ -M_{21} & +M_{22} & -M_{23} \\ +M_{31} & -M_{32} &+M_{33} \end{bmatrix}\)

\(\ Cofactor \ Matrix = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{bmatrix}\)

For the cofactor of a matrix, few steps needed are as follows:

• Step 1: First, find the minor of each element to form a minor matrix.

• Step 2: To find the cofactor, multiply each minor by (-1)^{(i + j)}.

• Step 3: Use these cofactors to create a matrix.

Let’s learn with an example:

Determine the cofactor matrix.

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

• Step 1: Finding the minors
   

1. M₁₁ = det \(\begin{bmatrix} 5& 6 \\ 8 & 9\end{bmatrix}\)

   M₁₁ = \((5 × 6) - ( 6 × 8) = -3\) 

    

2. M₁₂ = det \( \begin{bmatrix} 4& 6 \\ 7 & 9\end{bmatrix}\)

   \(M₁₂ = (4 × 9) - ( 6 × 7) = -6\)
    
3. Repeat these steps for every element in the matrix.
   
   Minor matrix: \(\begin{bmatrix} -3 & -6 & -3 \\ -6 & -12 & -6 \\ -3 & -6 & -3\end{bmatrix}\)
    

• Step 2: Finding the Cofactor matrix:
  Multiplying each minor by (-1)^{(i + j)}
  
  Cofactor matrix: \(\begin{bmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ 3 & 6 & -3\end{bmatrix} \)

Cofactor of a 2 𝗑 2 matrix:

If A = \( \begin{bmatrix} a& b \\ c & d\end{bmatrix}\)

So, the cofactor matrix is: \( \begin{bmatrix} d & -b \\ -c & a\end{bmatrix}\)

Cofactor of a 3 𝗑 3 matrix:

\(A = \begin{bmatrix} a& b & c\\ d & e & f \\ g & h & i\end{bmatrix}\)
 

1. To find the cofactor matrix of A, we need to determine the cofactor for each element one by one.
   
   Here, M_ij represents the minor of the element in row i and column j,
   
   \(M_{11} = \begin{vmatrix} e & f \\h & i \end{vmatrix}\\ \space \\M_{12} = \begin{vmatrix} d & f \\ g& i \end{vmatrix}\)
    
2. Calculating the minor of each element in the matrix results in the following matrix:
   
   \(M = \begin{bmatrix} M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} &M_{23} \\ M_{31} & M_{32} &M_{33} \end{bmatrix}\)
    
3. Now, we calculate the cofactor of each element by multiplying the corresponding minor from the minor matrix with (-1)^{i +j}, where i and j are the row and column numbers of that element.
   
   C₁₁ = (-1)^{1 + 1} ⋅ M₁₁ = M₁₁
   C₁₂ = (-1)^{1 + 2} ⋅ M₁₂ = - M₁₂
   C₁₃ =  (-1)^{1 + 3} ⋅ M₁₃ = M₁₃
    
4. Repeat the steps to calculate each cofactor to obtain the cofactor matrix:
   
   \(\ Cofactor \ Matrix = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{bmatrix}\)

The cofactor matrix plays an important role for calculating the determinant of a matrix. Let’s look at each of them:

 

• Determinant of a Matrix
   

1. For calculating the determinant of a matrix, choose any column or row from the matrix.
    
2. Multiply each element in column or row by its corresponding cofactor.
    
3. Finally, add all these values to get a determinant. 
   
   \(A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} &a_{23} \\ a_{31} & a_{32} &a_{33} \end{bmatrix}\)
   
   For matrix A, the determinant is represented by |A|: 
   |A| = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃
   
   Here: 

• C_ij is the cofactor of the element a_ij, which can be calculated by:
  \(Cᵢⱼ = (–1)⁽ⁱ\ ⁺\ ʲ⁾ × Mᵢⱼ\)
  
  Where:
• M_ij represents the minor of a_ij, which is the determinant of the (n – 1) × (n – 1) matrix formed by removing the i^th row and j^th column from matrix A.
  
   
• Adjoint of a Matrix
  
  We determine the adjoint of a matrix using the following steps:
   

1. We first calculate the cofactor matrix:
   
   \(\ Cofactor \ Matrix = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{bmatrix}\)
    
2. Then take the transpose of the cofactor matrix to obtain the adjoint:
   
   \(Adj (A) = \begin{bmatrix} C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32} \\ C_{13} & C_{23} & C_{33} \end{bmatrix}\)

• Inverse of a Matrix
  
  When |A| is not equal to 0, the inverse of a matrix A will be represented as:
  
  \(A^{-1} = {1 \over{ |A|} } \times  { Adj (A)}\)
  
  To find the inverse:

1. We first find the determinant.
    
2. Then, determine the adjoint.
    
3. Now, divide each element of the adjoint by the determinant.

Students from small grades can face difficulties when solving cofactor matrix. To make the process easy and efficient, here are some common tips:

1. Use the sign pattern to find the correct signs when calculating cofactor matrix.
    
2. To find the adjoint, take the transpose of the cofactor matrix.
    
3. Remember the relationship: \(A^{-1} = {1 \over{ |A|} } \times  { Adj (A)}\), to find the inverse.
    
4. To find the adjoint of a 2 × 2, simply interchange the non-diagonal elements and change the sign of the diagonals elements.
    
5. First practice 2 × 2 matrix to understand the process, and then move to the 3 × 3 matrix.
    

Parents Tips: Encourage your child to practice by solving different problems. You can also use a cofactor matrix calculator to check your children's calculation and verify final answers.

As we have learned, the cofactor matrix plays a vital role in various mathematical operations, such as finding the inverse of a matrix. We will now look at how it can be applied in real-life situations beyond mathematics. 

• In physics and engineering, people use Cramer’s rule to solve systems of equations. It is also used to solve equations for current and voltages in electrical circuits.

• The cofactor matrix is widely applied in cryptography for coding and decoding secret messages. Matrices are also used to program the face recognition to AI.

• Businesses utilize cofactor matrices in mathematical models to analyze and forecast pricing strategies, market demand, and sales trends, helping them make better decisions.

• In computer animations and video games, the turning, jumping of a character or throwing of an item is programmed using matrices and their cofactors.

• To located your position using GPS, computers use algorithms based on matrix calculation. Matrix calculation are also used to track satellites, guide spacecraft in outer space.