What Are The Cofactors?

The concept of the cofactor is applicable to a matrix which is an array of numbers or elements arranged in rows and columns. Each element in the matrix can be identified by its position corresponding to the particular row and column in which it lies. The cofactor for any element is obtained by first forming a matrix taking elements from the original matrix but excluding elements from the row and column holding that particular element. The next step is to find the determinant of this new smaller matrix to get a value which is known as the minor of that element. This article very well explains the difference between minors and cofactors.

Steps to Find Minor of an Element 

The cofactor of any element of a matrix is determined using certain steps, which are described as follows: 

1. Identify the element positions in terms of the row and column it lies.

2. Form a new matrix by deleting the particular row and column elements.

3. Calculate the determinant of the new matrix to get the value of the minor of that element.

4. Take the row and column number corresponding to that element and add the values.

5. This sum is the exponent for (-1) and finds the value.

6. Multiply the value obtained in the above step (which is either +1 or -1) with the nominal value obtained in step 4 to get the value of the cofactor of that element.

Concept of Cofactor and Minor

The minor of a particular matrix element is determined by taking the determinant of a new matrix formed after excluding elements that belong to the particular row and column in which that element belongs. The cofactor of an element is determined by the process of multiplication of two numerical values, one of which is minor and the other is the value of (-1) raised to the power of the sum of values indicating the row and column for that particular element in the original matrix. The numerical values may be the same at times, but there are basic differences between minor and cofactor. 

Difference Between Minor and Cofactor

The differences between minor and cofactor are observed by their calculation process. The minor is obtained by taking the determinant of the matrix formed by selected elements of the original matrix. The cofactor is the value of the product of minor and (-1) raised to the power of the sum of the values representing the row and column respectively of that element.

The difference between the minor and cofactor of an element in a matrix can be explained with the following example. 

Let’s consider a square matrix having three rows and three columns. We have to find the minor and cofactor of a 3 x 3 matrix for an element in the second column position of the third row. 

• Finding the minor: The minor of that element can be determined by forming a matrix excluding all elements of the second column and third row of the given matrix. Now the determinant of this matrix gives the nominal value for that particular element. 

• Finding the cofactor: The element’s cofactor is determined by multiplying the value of minor as obtained in the above step by (-1) raised to the power of 5 (=3 + 2) because 3 and 2 are the row and column numbers, respectively for that element. 

The difference between the minor and cofactor of an element also lies in the significant difference as they can have the same or different signs depending on the row and column values. The sum of these values becomes the exponent of (-1), which makes the value either +1 or -1. Accordingly, the minor and cofactor will have the same or opposite sign.

Finding Minor and Cofactor

Let’s go through the process to find the minor and cofactor of a 3 x 3 matrix. This matrix contains nine elements arranged in three rows and three columns. 

Suppose we have to find the minor and cofactor of a 3 x 3 matrix for the element lying in the first row of the second column. The steps to be followed are described below.

1. The positional value of the element indicates row number 1 and column number 2.

2. All the elements in row 1 and column 2 of the matrix are excluded.

3. A new matrix is formed by taking the remaining four elements. This matrix is a 2 x 2 matrix.

4. The determinant of this new matrix is calculated by multiplying the diagonal elements and then subtracting the second value from the first.

5. The row value of 1 and column value of 2 are added to get the three (= 1 + 2).

6. The value of (-1) raised to the exponent value of 3 is calculated (-1).

7. The value obtained in step 4 is minor, multiplied by the value obtained in step 6 to get the element’s cofactor.

Conclusion

In short, a cofactor in a number is obtained by removing the rows and columns of a particular element given in the square or rectangular form of the matrix. Hope you have understood the information given in this article curated by our experts. To get more information on cofactors, visit our website. If you want to learn about the difference between cofactors or minors, the abovementioned information and differences will help you know more about these terms. It will help you increase your knowledge and clear all the terms.