Minors and Cofactors of Determinants

Matrix is an array of real numbers(or other suitable entities), arranged in rows and columns where entities are referred to elements present in the Matrix. Below image demonstrates the Matrix, where elements separated horizontally are known as rows of the Matrix and the elements separated vertically are know as columns of the Matrix.

As we know Matrix is arranged in rows and columns, the below matrix having 3 rows and 3 columns, So the order of the Matrix is 3 × 3.

Any four elements a, b, c, and d are arranged in two rows and two columns between two Vertical bars as shown below, forms which are called Determinant of the second-order or second-order determinant. As shown below demonstrating the Determinant and expansion of Determinant.

Determinant of a Matrix

Determinant is useful for solving linear equations, capturing how linear transformation change area or volume, and changing variables in integrals. The determinant can be viewed as a function whose input is a square matrix and whose output is a number. In the below article we are discussing the Minors and Cofactors thoroughly. In simple language we can say, To every small matrix A, we can associate a number (real or complex) which is called the determinant of a square matrix A. 

Determinant of a matrix can be easily represented as det (A) or | A |

Now let’s jump to our topic which is Minors and Co-factors. 

So firstly let’s discuss the Minors.

Note:

• The questions which are done in this article have appeared in different previous year question papers.
• i represent rows of the determinant whereas j represents columns of the determinant.
• I am highlighting the ij^th terms so that you can see clearly without any confusion.
• In the below article you will see the questions and solution of that question is demonstrated by the image.

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

Steps for Computing Minor of a 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.

Sample Problems on Minor of a Matrix

Problem 1: If the matrix A is

then, write 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.

Now after deleting i^th row and j^th column we had to expand the determinant, so we get (8 – 15) which on solving gives -7, which is our required answer.

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

Problem 2: If the matrix A is

then find out minor of a_32.

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:

By following the same procedure as of the above question we had solved this question too through the expansion of determinant as we discussed in our Introduction.

Co-factors of a Matrix

Co-factor of an element a_ij of a determinant, denoted by A_ij or C_ij , is defined as A_ij = (-1)^i+j M_ij , where M_ij is a minor of an element a_ij

Formula to find cofactors 

A_ij = (-1)^i+j  M_ij 

Sample Problems on Co-factors of a Matrix

Problem 1: If a matrix A is

write the cofactor of the element a_32.

Solution:

As asked in question we have to find the co factor of element a32 which means our row (i) = 3 and column (j) = 2 so we have row and column as we do to find the minor by deleting the rows and column at which asked element exist we do the same in this question to and then put that in our formula -> A_ij = (-1)^i+j  M_ij

So after putting in the formula of finding cofactor and doing expansion of determinant we get (-1) (5 – 16) which on solving gives the answer 11, this is our required answer.

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

Solution: 

In the question, we are having determinant. So we have row and column given in the question.

Here, a₃₂ = 3+2 = 5

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.

So, 110 is our required answer.