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.

The minor of the element a₁₂ is:

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

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:

Step 1: Finding the minors
M₁₁ = det

M₁₂ = det

Repeat these steps for every element in the matrix.
Minor matrix:

Multiplying each minor by (-1)^{(i + j)}
Cofactor matrix:

Cofactor of a 2 𝗑 2 matrix:
If A = 

So, the cofactor matrix is:

Cofactor of a 3 𝗑 3 matrix:
 
To find the cofactor matrix of A, we need to determine the cofactor for each element one by one.
Determine the cofactors of each of its elements:

Here, M_ij represents the minor of the element in row i and column j,

Calculating the matrix of each element in the matrix results in the following matrix:

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

Repeat the steps to calculate each cofactor to obtain the cofactor matrix:

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

For calculating the determinant of a matrix, choose any column or row from the matrix. Multiply each element in column or row by its corresponding cofactor. Finally, add all these values to get a determinant. 
 
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:

C_ij = (-1)^i+j ⋅ M_ij

Where:

M_ij represents the minor of a_ij, which is the determinant of the (n –1) × (n –1) matrix formed by removing the ith row and jth column from matrix A.

Adjoint of a Matrix

We determine the adjoint of a matrix using the following steps:
Step 1: We first calculate the cofactor matrix:

 
Now, let’s take the transpose of the cofactor matrix to obtain the adjoint:

Inverse of a Matrix

When |A| is not equal to 0, the inverse of a matrix A will be represented as:

A-1 = 1/ |A|  ⋅ Adj (A)

To find the inverse:

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

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.

• The cofactor matrix is widely applied in cryptography for coding and decoding secret messages.

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