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 \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}is calculated as:

M_{11} = det\begin{bmatrix} 5 & 6\\ 8 & 9 \end{bmatrix}\\ = 45-48\\ = -3

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

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 \begin{bmatrix} 5 & 6\\ 8 & 9 \end{bmatrix}
M₁₁ = 5(9) - 6(8)
M₁₁ = 45 - 48 = -3

Similarly, the minor of element a₁₂ is calculated:

M₁₂ = determinant of \begin{bmatrix} 4 & 6\\ 7 & 9 \end{bmatrix}
M₁₂ = 4(9) - 6(7)
M₁₂ = 36 - 42 = -6

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

\begin{bmatrix} -3 & -6 & -3\\ -6 & -12 & -6\\ -3 & -6 & -3 \end{bmatrix}

• 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:

\begin{bmatrix} -3 & 6 & -3\\ 6 & -12 & 6\\ -3 & 6 & -3 \end{bmatrix}

Cofactor of 2×2 Matrix

Consider a 2×2 matrix as follows: C = \begin{bmatrix} a & b\\ c & d \end{bmatrix}

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

C = \begin{bmatrix} a & -b\\ -c & d \end{bmatrix}

Cofactor of 3×3 Matrix

Consider a 3×3 matrix as follows: A = \begin{bmatrix} a & b & c\\ d & e&f\\ g&h&i \end{bmatrix}

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:

M_{11} = det \begin{bmatrix} e&f\\ h&i \end{bmatrix}
M_{12} = det \begin{bmatrix} d&f\\ g&i \end{bmatrix}

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

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

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:

C= \begin{bmatrix} C_{11} & C_{12} & C_{13}\\ C_{21} & C_{22} & C_{23}\\ C_{31} & C_{32} & C_{33} \end{bmatrix}

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 \begin{bmatrix} 1 & 3\\ 5 & 7 \end{bmatrix}

Given matrix is \begin{bmatrix} 1 & 3\\ 5 & 7 \end{bmatrix}
Minor M₁₁ = 7
Cofactor of a₁₁ = 7 × (-1)¹⁺¹ = 7

Example 2. Find the cofactor of a₁₂ in the matrix \begin{bmatrix} 1&4 & 6\\ 8 & 7 & 9 \\ 4 & 0 & 5 \end{bmatrix}

Given matrix is \begin{bmatrix} 1&4 & 6\\ 8 & 7 & 9 \\ 4 & 0 & 5 \end{bmatrix}
Minor M₁₂ = determinant of \begin{bmatrix} 8 & 9 \\ 4 & 5 \end{bmatrix}
M₁ = 40 - 36 = 4
Cofactor C₁ of a₁₂ = M₁₂ × (-1)¹⁺²
C₁₂ = 4 × (-1) = -4

Example 3. What is the cofactor matrix of \begin{bmatrix} 3 & 6\\ 7 & 8 \end{bmatrix}

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:
\begin{bmatrix} 8 & 7\\ 6 & 3 \end{bmatrix}

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:
\begin{bmatrix} 8 & -7\\ -6 & 3 \end{bmatrix}

Example 4. What is the cofactor matrix of \begin{bmatrix} 0 & -6\\ -3 & -4 \end{bmatrix}

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:
\begin{bmatrix} -4 & -3\\ -6 & 0 \end{bmatrix}

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:
\begin{bmatrix} -4 & 3\\ 6 & 0 \end{bmatrix}

Example 5. What is the cofactor matrix of \begin{bmatrix} 2&4 & 6\\ 7 & 9&2\\ 1& 6 & 0 \end{bmatrix}

Step 1: Find the minor of each element and make a minor matrix.
M₁₁ = determinant of \begin{bmatrix} 9 & 2\\ 6 & 0 \end{bmatrix}
M₁₁ = 0 -12 = -12
M₁₂ = determinant of \begin{bmatrix} 7 & 2\\ 1 & 0 \end{bmatrix}
M₁₂ = 0 - 2 = -2
Similarly calculate minors of all elements to obtain the following minor matrix:
\begin{bmatrix} -12 & -2 & -33\\ -36 & -6 & 8\\ -46 & -38 & -10 \end{bmatrix}

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:
\begin{bmatrix} -12 & 2 & -33\\ 36 & -6 & -8\\ -46 & 38 & -10 \end{bmatrix}

Practice Problems on Cofactor of a Matrix

Problem 1: Find the cofactor of the element in the second row and third column: A = \begin{bmatrix} 2 & 3 & 4\\ 6 &6 & 5\\ 7 & 8 & -10 \end{bmatrix}.

Problem 2: Find the cofactor matrix of A = \begin{bmatrix} 12 & 7 & 4\\ 4 &1 & 5\\ 7 & 0 & 5 \end{bmatrix}.

Problem 3: Find the cofactor matrix of A = \begin{bmatrix} 4 & 0 & 1\\ 16 &2 & 9\\ 8 & 3 & 7 \end{bmatrix}.

Problem 4: Find the cofactor matrix of A = \begin{bmatrix} 1 & 0 & 0\\ 1 &0 &1\\ 4& 1 & 6 \end{bmatrix}.

Problem 5: Find the cofactor matrix of A = \begin{bmatrix} 3 & 5 & 7\\ 9 &11 & -13\\ -17 & 19 & 23 \end{bmatrix}.

Solutions of Practice Problems:

1. \begin{bmatrix}-100 & 95 & 6 \\62 & -48 & 5 \\-9 & 14 & -6\end{bmatrix}
2. \begin{bmatrix} 5 & 15 & -7\\ -35 & 32 & 49\\ 31 & -44 & -16 \end{bmatrix}
3. \begin{bmatrix} -13 & -40 & 32\\ 3 &20 & -12\\ -2 & -20 & 8 \end{bmatrix}
4. \begin{bmatrix} -1 & -2 & 1\\ 0 & 6 & -1\\ 0 & -1 & 0 \end{bmatrix}
5. \begin{bmatrix} 500 & 14 & 358\\ 18 &188 & -142\\ -142 & 102 & -12 \end{bmatrix}