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Table of contents
1.
Introduction
2.
Adjoint of a Matrix
3.
Inverse Matrix
4.
Difference between Adjoint and Inverse Matrices
5.
FAQs
6.
Key Takeaways
Last Updated: Mar 27, 2024

Adjoint and Inverse of a Matrix

Author Juhi Sinha
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23 Jul, 2024 @ 01:30 PM

Introduction

The adjoint of a matrix is the transpose of the cofactor of the matrix. The Adjoint of a Matrix can be used to calculate the inverse of a matrix easily.

In this article, we will discuss the adjoint and inverse of a matrix in detail. So without any further ado, let's start!

Adjoint of a Matrix

The adjoint of a matrix is the matrix obtained by taking the transpose of the cofactor matrix of a given square matrix. Adj represents the adjoint of any square matrix.

A cofactor is a number obtained by ignoring a specific element's row and column in the form of a square or rectangle. 

The adjoint of a matrix can be calculated using the cofactor and transpose of a matrix. It is denoted by adj A.                            

                           adj(A) = (Cofactor of Matrix A)T

Note: We can find the adjoint of the square matrix only.

Let us understand the adjoint of a matrix with an example:

 Q) Find the Adjoint of a matrix A given below:

First, we have to calculate all the cofactors of elements in matrix A.

 Cofactor of A11  i.e. 4 =  

= 1 X 4 - 6 X 2

= -8

Cofactor of A12  i.e. -2 =  

= 3 X 4 - 5 X 2

= 2

Cofactor of A13  i.e. 1 =  

= 3 X 6 - 5 X 1

= 13

Cofactor of A21 i.e. 3 =  

= -2 X 4 - 6 X 1

= -14

Cofactor of A22 i.e. 1 =  

= 4 X 4 - 5 X 1

= 11

Cofactor of A23 i.e. 2 =  

= 4 X 6 - (-2) X 5

= 34

Cofactor of A31 i.e. 5 =  

= -2 X 2 -  1 X 1

= -5

Cofactor of A32 i.e. 6 =  

=  4 X 2 -  3 X 1

=  5

Cofactor of A33 i.e. 4 =  

=  4 X 1 -  3 X (-2)

=  10

Now, we have to multiply the cofactor of elements with + or - using the formula (-1)i+j . If i+j is even, then we have to multiply it with positive (+) else multiply it with negative (-) sign.

The sign convention for a 3 X 3 matrix is given below: 

Now, we will multiply the cofactors with the given sign convention: 

After multiplying with the sign, the matrix obtained is:

Now transpose of matrix A is given below:

As we know, the transpose of a matrix's cofactor is its adjoint, so the above matrix is the required answer.

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Inverse Matrix

When we multiply two matrices together, the inverse of a matrix is defined as the identity matrix. If XY = YX = I, Y is the inverse matrix of X, and X is the inverse matrix of Y.

The fact that A's determinant is not zero is a necessary and sufficient condition for a matrix to be invertible. In other words, |A| = det(A)  ≠  0. If a matrix meets this condition, it is said to be invertible, non-singular, or non-degenerative. As a result, A is a square matrix, and A-1 and A are of equal size.

If matrix A is invertible, its inverse can be obtained by the formula given below:                          

                              

Let us understand the inverse of a matrix using the above formula with an example:

Q) Find the inverse of a matrix given below:

  

Using the above method to calculate adjoint, we get the adjoint matrix as:

 

Now as per the above formula,

     

|A|= -8(11 X10 - (-34) X (-5)) - 14( 2 X 10 - 13 X (-5)) + (-5)(2 X -34 - 13 X 11)

    =  480 - 1190 + 1055

    =  345

We will get A-1   as:   

   

So, the above matrix is the required inverse matrix.

Difference between Adjoint and Inverse Matrices

The differences between Adjoint of matrix and inverse of a matrix are as follows:

• A matrix's adjoint is the transpose of the cofactor matrix, whereas an inverse matrix is a matrix that gives the identity matrix when multiplied together.

• The adjoint matrix is one of the most common methods of manually finding the inverses and can be used to calculate the inverse matrix.

• An adjoint matrix exists for every matrix, but the inverse only exists if the determinant is non-zero.

Check out this problem - Matrix Median

FAQs

1. What is the difference between an adjoint and inverse matrix?

A matrix's adjoint is the transpose of the cofactor matrix, whereas an inverse matrix is a matrix that gives the identity matrix when multiplied together.

2. Can we multiply two matrices with different sizes?

Yes, we can multiply two different sizes of matrices, but they must follow the condition that the number of columns of the first matrix should be equal to the number of rows of the second column.

3. What is a cofactor of a matrix?

A cofactor is a number obtained by ignoring a specific element's row and column in the form of a square or rectangle. 

4. What is the inverse of a matrix?

When we multiply two matrices together, the inverse of a matrix is defined as the identity matrix. If XY = YX = I, Y is the inverse matrix of X, and X is the inverse matrix of Y.

Key Takeaways

In this article, we have extensively discussed the adjoint of a matrix, the calculation of an inverse using adjoint with the help of examples to understand it better.

We hope that this blog has helped you enhance your knowledge regarding matrix and if you would like to learn more, check out our articles on Matrix and Types of Matrices. Do upvote our blog to help other ninjas grow.

 “Happy Coding!”

 

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