Minors and Cofactors

Minors and cofactors play a crucial role in the calculation of the adjoint and inverse of a matrix. They are essential for determining the determinant and are widely used in various applications, including computer science.

Minors:

A Minor is a smaller part of the larger matrix obtained for a particular element of the matrix by deleting the terms of the row and column to which the element belongs.

In simpler terms:

1. Choose an element of the matrix.
2. Remove the row and column containing that element.
3. The minor is the determinant of the remaining matrix.

Cofactors:

A cofactor is closely related to the minor.

• It is calculated by multiplying the minor by (−1)^i+j, where i and j are the row and column indices of the element.
• The sign factor (−1)^i+j accounts for the position of the element within the matrix.

Minor of a Matrix

Minor of an element a_ij of a determinant is a determinant obtained by deleting the i^th row and j^th column in which element a_ij lies. Minor of an element a_ij is denoted by M_ij.a

How to Find a Minor of a Matrix?

Use the following steps to find the minor of any given matrix:

Step 1: Hide the i^th row and j^th column of the matrix A, where the element a_ij lies. 

Step 2: Now compute the determinant of the matrix after the row and column is removed using step 1.

Step 3: Result of Step 2 is the minor for he elelment of the i^th row and j^th column, then repeat the process for each element of the matrix to find the minor for all the elements of the matrix.

Sample Problems on Minor of a Matrix

Problem 1: If the matrix \bold{A = \begin{bmatrix} 1&2&3\\ 2&0&1\\ 5&3&6 \end{bmatrix}}then find the minor of  a_22.

Solution:

In this question, we have to find out the minor of a₂₂, the element present at a₂₂ is 0. As we learn from our definition of a minor we have to delete the i^th row and j^th columns at which our asked element is present. Below image is demonstrating how to delete the i^th row and j^th column

After deletion, we write our left element as it is and do cross multiplication.

Minor of element a₂₂ = \begin{vmatrix} 1&3\\ 5& 6 \end{vmatrix}

⇒ Minor of element a₂₂ = 6 - 15 = -9

Note: Always remember, after the multiplication of the left diagonal element in the second-dimensional determinant, always put the -ve sign then do the multiplication of right diagonal elements and solve them out.

Problem 2: If the matrix \bold{A = \begin{bmatrix} 1 & 0 & 4\\ 3 & 5 &-1\\ 0 & 1 & 2 \end{bmatrix}}, then find the minor of a₃₂.

Solution:

In the above question we have asked to find out the minor of a₃₂ element which is 1. So as we did in this above problem same procedure we will follow. So firstly we have to delete the i^th row and j^th column at which our element is present.

So we had canceled the i^th row and j^th column at which our element is present. So write the elements which are left as it is.

Then do the cross multiplication and solve:

Minor of element a₃₂ = \begin{vmatrix} 1&4\\ 3& -1 \end{vmatrix}

Minor of element a₃₂ = -1 -12 = -13

Cofactor of a Matrix

Co-factor of an element a_ij of a determinant, denoted by A_ij or C_ij, and is defined as follows:

C_ij = (-1)^i+j M_ij 

Where,

• M_ij is the minor of the element a_ij, and 
• i and j respectively reprensent the number of row and column of the element (position).

How to Find Cofactors of a Matrix

Let's go step-by-step with an example to understand how to find the cofactor of a matrix.

Example 1: Given the matrix A

\bold{|A| = \begin{vmatrix} 5&3&1\\ 2&0&-1\\1&2&3 \end{vmatrix}}

Find the cofactor of the element a_32.

Solution:

Step 1: Identify the Position of the Element

• The element a₃₂ is located at the 3rd row and 2nd column.
• To find the cofactor, we need to calculate the minor of the matrix after removing the 3rd row and 2nd column.

Step 2: Find the Minor (M_ij)

• The minor is the determinant of the matrix that remains after deleting the row and column that contain the element a₃₂.
• After removing the 3rd row and 2nd column, we are left with the matrix:
  • \bold{|A| = \begin{vmatrix} 5&1\\2&-1\end{vmatrix}}
• Now, we calculate the determinant of this 2x2 matrix:
  • M_{32} = \begin{vmatrix} 5&1\\2&-1\end{vmatrix} = (5 × −1) − (1 × 2) = −5 −2 = −7

Step 3: Apply the Formula for Cofactor

• The cofactor of an element is given by the formula:
  • Cij = (−1)^i+j × M_ij
• For the element a₃₂, we have i = 3 and j = 2, so:
  • C₃₂ ​= (−1)³⁺² × M₃₂ ​ =(−1)⁵ × (−7) = −1 × (−7) = 7

Example 2: If the A_ij of the element a_ij of the determinant is given below, then write the value of a₃₂. A₃₂.

 \bold{|A| = \begin{vmatrix} 2&-3&5\\ 6&0&4\\1&5&-7 \end{vmatrix}}

Solution: 

Similary in this question, we are having determinant. So we have row and column given in the question.
Here, a₃₂ = 5 (second element of third row)

Given, A_ij is the cofactor of the element a_ij of A . So now we can solve this question by putting the values in the formula of cofactor as discussed in above question.

Cofactor of elements (A₃₂)= (-1)^{3+2}\begin{vmatrix}2 &5\\6&4 \end{vmatrix}

⇒ Cofactor of element (A₃₂) = (-1)(8 - 30) = 22

 a₃₂ . A₃₂ = 5 . 22 = 110

So, 110 is our required answer.

Application of Minors and Cofactors

Minors and Cofactors are used in the calculation of the following terms:

• Adjoint of Matrix
• Inverse of Matrix

Adjoint of Matrix

To calculate the adjoint of Matrix, you need to follow the following steps:

Step 1: Calculate the cofactors of each element of a given matrix.
Step 2: Construct the matrix from the cofactor of elements.
Step 3: Calculate the Transpose of the resultant matrix in Step 2.
Step 4: Resulting matrix of Step 3 is the adjoint of the given matrix.

Example: Find the adjoint of the following matrix A;

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

Solution:

Step 1: Compute the cofactors of each element in A.

C₁₁ = 5 × 9 - 6 × 8 = -3
C₁₂ = -(4 × 9 - 6 × 7) = 6
C₁₃ = 4 × 8 - 5 × 7 = -3
C₂₁ = -(2 × 9 - 3 × 8) = 6
C₂₂ = 1 × 9 - 3 × 7 = -6
C₂₃ = -(1 × 8 - 2 × 7) = 3
C₃₁ = 2 × 6 - 3 × 5 = -3
C₃₂ = -(1 × 6 - 3 × 4) = 6
C₃₃ = 1 × 5 - 2 × 4 = -3

Step 2: Construct the matrix of cofactors.

Matrix of cofactors, C = \begin{bmatrix} -3&6&-3\\ 6&-6&3\\ -3&6&-3 \end{bmatrix}

Step 3: Transpose the matrix of cofactors.

C' = \begin{bmatrix} -3&6&-3\\ 6&-6&6\\ -3&3&-3 \end{bmatrix}

Step 4: The resulting matrix is the adjoint of A.

adj(A) = C' = \begin{bmatrix} -3&6&-3\\ 6&-6&6\\ -3&3&-3 \end{bmatrix}

Inverse of Matrix

To calculate the inverse of a matrix, you can use the following steps:

Step 1: Calculate the determinant of the given matrix.

Step 2: If the value of the determinant is zero, then the matrix has no inverse. Otherwise, calculate the adjoint of the matrix using the steps mentioned in the previous heading.

Step 3: Use the formula A^{-1} = \frac{adj(A)}{|A|}, to calculate the inverse of the given matrix.

Example: Find the inverse of the following matrix A:

\bold{A = \begin{bmatrix} 2&-1&0\\ -1 &2 &-1\\ 0&-1&2 \end{bmatrix}}

Solution:

Step 1: Find the determinant of A.

det(A) = 2(2 × 2 - (-1) × (-1)) - (-1)((-1) × 2 - (-1) × 0) + 0 

⇒ det(A) = 6 - 2 = 4

Step 2: Calculate the adjoint of A using the steps mentioned in the previous example.

C₁₁ = 4 - 1 = 3
C₁₂ = -(-2 - 0) = 2
C₁₃ = 1 - 0 = 1
C₂₁ = -(-2 - 0) = 2
C₂₂ = 4 - 0 = 4
C₂₃ = -(-2 - 0) = 2
C₃₁ = (1 - 0) = 1
C₃₂ = -(-2 - 0) = 2
C₃₃ = 4 - 1 = 3

Matrix of cofactors,

C = \begin{bmatrix} 3&2&1\\ 2&4&2\\ 1&2&3 \end{bmatrix}

Transpose of matrix of cofactors,

adj(A) = C' =  \begin{bmatrix} 3&2&1\\ 2&4&2\\ 1&2&3 \end{bmatrix}

Step 3: Multiply the adjoint of A by the reciprocal of the determinant.

 A^{-1} = \frac{adj(A)}{4} = \frac{1}{4} \times\begin{bmatrix} 3&2&1\\ 2&4&2\\ 1&2&3 \end{bmatrix}

\Rightarrow A^{-1}= \begin{bmatrix} \frac{3}{4}&\frac{2}{4}&\frac{1}{4}\\ \frac{2}{4}&\frac{4}{4}&\frac{2}{4}\\ \frac{1}{4}&\frac{2}{4}&\frac{3}{4} \end{bmatrix}

\Rightarrow A^{-1}= \begin{bmatrix} \frac{3}{4}&\frac{1}{2}&\frac{1}{4}\\ \frac{1}{2}&1&\frac{1}{2}\\ \frac{1}{4}&\frac{1}{2}&\frac{3}{4} \end{bmatrix}

Also Check

• Elementary Operation on Matrices
• Order of Matrix