Cofactor of a Matrix: Formula and Examples

A cofactor of a matrix is an important concept in linear algebra, often used in the calculation of determinants and inverses of matrices. The cofactor of an element in a matrix is computed as follows:

1. Minor: For a given element a_ij​ in a matrix, first find theminor of that element. The minor of a_ij​, denoted as M_ij​, is the determinant of the matrix that remains after removing the i-th row and j-th column from the original matrix.
2. Cofactor: The cofactor of a_ij​, denoted as C_ij​, is then obtained by multiplying the minor M_ij​ by (−1)^i+j. This factor (−1)^i+j accounts for the sign change that depends on the position of the element.

The cofactors for the above matrix is given below:

• C₁₁= (-1)² (3 - 16) = -13
• C₂₁= (-1)³ (-1 - (-2)) = -1

For example minor of the element a₁₁ matrix is calculated as:

Formula for Cofactor of a Matrix

If we denote the Cofactor using C_ij, then the cofactor of any element for 

C_ij = M_ij × (-1)^i+j

Where,

• i is the number of rows for the element under consideration,
• j is the number of columns for the element under consideration, and
• M_ij is the minor of the element in the i^th row and j^th column.

How to Find Cofactor of a Matrix?

In order to find a cofactor matrix we need to perform the following steps:

• Step 1: Find the minor of each element of the matrix and make a minor matrix.
• Step 2: Multiply each element in the minor matrix by (-1)^i+j_.
  Thus, we obtain the cofactor matrix.

Let us understand how to find a cofactor matrix using an example:

Example: Find the cofactor matrix of 

Solution:

• Step 1: Find the minor of each element and make a minor matrix.

Minor of a₁₁ is calculated by eliminating row 1 and column 1 and taking the determinant of the remaining matrix as follows:

M₁₁= determinant of 
M₁₁ = 5(9) - 6(8)
M₁₁ = 45 - 48 = -3

Similarly, the minor of element a₁₂ is calculated:

M₁₂ = determinant of 
M₁₂ = 4(9) - 6(7)
M₁₂ = 36 - 42 = -6

• Similarly, calculate the minors of all elements to obtain the following minor matrix:

• Step 2: Multiply each element of the minor matrix by (-1)^i+j to get the cofactor of that element i.e. C_ij

C₁₁ = M₁₁ × (-1)¹⁺¹ = M₁₁ = -3
C₁₂ = M₁₂ × (-1)¹⁺² = -M₁₂ = 6

• Similarly, calculate the other cofactors to obtain the following cofactor matrix:

Cofactor of 2×2 Matrix

Consider a 2×2 matrix as follows:

Then the cofactor matrix of any such matrix is written as:

Cofactor of 3×3 Matrix

Consider a 3×3 matrix as follows:

Then the cofactor matrix of any such matrix is calculated by calculating the cofactors of each of the elements as follows:

Let M_ij denote the minor of the element in row i and column j, then in the above matrix:

Similarly, we can calculate the minors of all the elements to get the below minor matrix:

Now cofactor of each element is calculated by multiplying the minor from the minor matrix with -1 raised to the power of the sum of row and column numbers to which that minor belongs as follows:

Let C_ij denote the cofactor of minor M_ij, then:

C₁₁ = (-1)¹⁺¹M₁₁ = M₁₁
C₁₂ = (-1)¹⁺²M₁₂ = -M₁₂
C₁₃ = (-1)¹⁺³M₁₃ = M₁₃

Similarly, after calculating all the cofactors of each element we get the following cofactor matrix:

Applications of Cofactor of a Matrix

There are various applications of Cofactor Matrix. Some of these applications are:

• Cofactor of the Matrix is used to find the adjoint of the matrix.
• Cofactor Matrix is used in the calculation of determinant of the matrix.
• It is also used to find the inverse of the matrix.

Also Check:

• Types of Matrix
• Transpose of Matrix

Solved Examples on Cofactor of a Matrix

Example 1. Find the cofactor of a₁₁ in the matrix 

Given matrix is 
Minor M₁₁ = 7
Cofactor of a₁₁ = 7 × (-1)¹⁺¹ = 7

Example 2. Find the cofactor of a₁₂ in the matrix 

Given matrix is 
Minor M₁₂ = determinant of 
M₁ = 40 - 36 = 4
Cofactor C₁ of a₁₂ = M₁₂ × (-1)¹⁺²
C₁₂ = 4 × (-1) = -4

Example 3. What is the cofactor matrix of 

Step 1: Find the minor of each element and make a minor matrix.
Minor of a₁₁ is calculated by eliminating the row 1 and column 1 as follows
M₁₁ = 8
Similarly minor of element a₁₂ is calculated by eliminating the row 1 and column 2 as follows:
M₁₂ = 7
Similarly calculate minors of all elements to obtain the following minor matrix:

Step 2: Multiply each element of the minor matrix by (-1)^i+j to get the cofactor of that element i.e. C_ij
Cofactor of M₁₁ is calculated as follows:
C₁₁ = M₁₁ × (-1)¹⁺¹ = M₁₁ = 8
Cofactor of M₁₂ is calculated as follows:
C₁₂ = M₁₂ × (-1)¹⁺² = -M₁₂ = -7
Similarly calculate the other cofactors to obtain the following cofactor matrix:

Example 4. What is the cofactor matrix of 

Step 1: Find the minor of each element and make a minor matrix.
M₁₁ = -4
M₁₂ = -3
Similarly calculate minors of all elements to obtain the following minor matrix:

Step 2: Multiply each element of the minor matrix by (-1)^i+j to get the cofactor of that element i.e. C_ij
C₁₁ = M₁₁ × (-1)¹⁺¹ = M₁₁ = -4
C₁₂ = M₁₂ × (-1)¹⁺² = -M₁₂ = -3
Similarly calculate the other cofactors to obtain the following cofactor matrix:

Example 5. What is the cofactor matrix of 

Step 1: Find the minor of each element and make a minor matrix.
M₁₁ = determinant of 
M₁₁ = 0 -12 = -12
M₁₂ = determinant of 
M₁₂ = 0 - 2 = -2
Similarly calculate minors of all elements to obtain the following minor matrix:

Step2 : Multiply each element of the minor matrix by (-1)^i+j to get the cofactor of that element i.e. C_ij
C₁₁ = M₁₁ × (-1)¹⁺¹ = M₁₁ = -12
C₁₂ = M₁₂ × (-1)¹⁺² = -M₁₂ = 2
Similarly calculate the other cofactors to obtain the following cofactor matrix:

Practice Problems on Cofactor of a Matrix

Problem 1: Find the cofactor of the element in the second row and third column: .

Problem 2: Find the cofactor matrix of .

Problem 3: Find the cofactor matrix of .

Problem 4: Find the cofactor matrix of .

Problem 5: Find the cofactor matrix of .

Solutions of Practice Problems:

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