Adjoint of a matrix

Introduction to Adjoint of a Matrix

An adjoint of a matrix is an important concept in mathematics, especially in linear algebra. The word "adjoint" means related to a matrix. In Simple terms, an adjoint of a matrix is the transpose of the cofactor matrix.

To find the adjacent, we calculate the cofactor for all the elements of the first given matrix, arrange them into a matrix, and then transpose it. This process is mainly used on square matrices such as 2 × 2 or 3 × 3.

Adjoint is very useful because it helps to find the inverse matrix, solves linear equations, and is used in many applications of algebra. In this article, we learn the steps for calculating the adjoint of a matrix, with properties, rules, and examples solved for better understanding.

 

Table of Contents

• Adjoint of a Matrix Definition
• Adjoint of a Matrix Formula
• Adjoint of a Matrix 2 × 2
• Adjoint of a 3×3 matrix
• How to calculate the adjacent matrix
• Step-by-step Method to Find Adjoint
• Adjoint of a Matrix Properties
• Practice Questions
• FAQs on Adjoint of a Matrix

 

Adjoint of a Matrix Definition

Let  A = [a_ij]_n×n be a square matrix of order 𝑛. The adjoint of A is defined as the transpose of the cofactor matrix of A.

In other words, if C = [A_ij]_n×nis the Cofactor matrix of 𝐴, then the adjoint of A is written as:

adj(A) = C^T

Here,

• a_ij = Matrix elements 𝐴 in he 𝑖th row and 𝑗th Column.

• A_ijAij = elements cofactor 𝑎 𝑖𝑗.

• C^T = Transpose of cofactor matrix.

Thus, the adjoint to a matrix is just the transpose of the cofactor matrix. It is mostly used to find the inverse of a square matrix.

 

Adjoint of a Matrix Formula

The formula around a matrix is based on two steps: finding the cofactor matrix and then taking its transporting it. If A = [a_ij]_n×n, there is a square matrix, and A_ij is the cofactor is the element a_ij. So the adjoint of 𝐴 is:

Adj (A) = [A_ij]^T

For a 2 × 2 matrix, the formula becomes very simple. If

Then,

[ a   b ]
   A = [ c   d ]         

So the adjoint can easily be calculated for a small matrix, and the same process also works for a large square matrix.

 

Adjoint of a Matrix 2 × 2

Let 𝐴 be a 2 × 2 matrix given by:

         [ a₁₁   a₁₂ ]
   A = [ a₂₁   a₂₂ ]

Then, the adjoint of this matrix is : 

                   [ A₁₁   A₁₂  ]^T
   adj (A) = [  A₂₁   A₂₂ ]

Here,

• A₁₁  = Cofactor of a₁₁ 

• A₁₂ = Cofactor of a₁₂ 

• A₂₁ = Cofactor of a₂₁ 

• A₂₂ = Cofactor of a₂₂ 

Alternatively, for a 2 × 2 matrix, the adjoint can also be found in a simpler way:

• Interchange the elements a₁₁ and a₂₂.

• Change the signs of a₁₂ and a₂₁.

So, 

                  [ a₂₂   - a₁₂ ]
   adj (A) = [ - a₂₁   a₁₁ ]

This makes it very quick and easy to calculate the adjoint of a 2 × 2 matrix.

 

Adjoint of a 3×3 matrix

Consider a 3 × 3 matrix:

     [ a₁₁   a₁₂    a₁₃]
   A = [ a₂₁   a₂₂   a₂₃]    
     [ a₃₁   a₃₂    a₃₃]

This adjoint matrix is found by taking the transpose cofactor matrix of 𝐴.

                [ A₁₁   A₁₂    A₁₃]^T
   adj (A) = [ A₂₁   A₂₂   A₂₃]    
              [ A₃₁   A₃₂    A₃₃]

 Hers each A_ij is the cofactor sector of the element a_ij.

What is a minor?

The minor of an element in a matrix is the determinant of the smaller matrix that we get after deleting the row and column of the element.

For example, the minor of the element 𝑎 21 is called M₂₁. It is obtained by removing the 2nd row & 1st column from the given matrix and then finding the determinant of the remaining 2 × 2 

What is a cofactor sector?

A cofactor matrix is the minor of an element with a sign (+ or -) depending on its position. Formulas to find a cofactor sector of an element a_ij are :

C_ij= (- 1)^i=j det (M_ij)

• M_ij = Minor of the element a_ij

• (- 1)^i=j provides a proper sign (positive or negative) depending on the potion.

• So, the cofactor matrix is made using all these cofactors, and the transpose gives the adjoint of a 3 × 3 matrix.

 

How to calculate the adjacent matrix

We can also write a common formula to find the adjoint of a square matrix of order  𝑛 × 𝑛. Let 𝐴 be a square matrix of order 𝑛 × 𝑛; its adjoint is given by the transpose of the cofactor matrix.

If,

A = [ a_ij ]_n*n

Then this matrix is adjoint to:

adj (A) = [A_ij]^T

• Here, A₁₁ ,A₁₂  ,A₂₁ ,A₂₂....A_nn elements are cofactors a₁₁ ,a₁₂  ,a₂₁ ,a₂₂....a_nn

• So simple words, the adjoint of a matrix to just the transpose of the cofator matrix of that given square matrix.

Step-by-step Method to Find Adjoint:

1. Start with a square matrix (order 𝑛 ×  𝑛).

2. Find the minors of each element by deleting the row and column of the element.

3. Calculate the cofactors using the formula:

C_ij = (- 1)^i+j det (M_ij)

Where M_ij is he minor of element  a_ij.

4. Form the cofactor matrix by arranging all cofactors in their respective positions.

5. Take the transpose of this cofactor matrix 

6. The result is the adjoint of the matrix, written as adj(A)

 

Adjoint of a Matrix Properties

Some important properties of adjoint matrices are given below. These properties are very useful when solving problems in matrices and determinants.

Let A be a square matrix of order n. Then:

1. A.(adj A) = (adj A). A = |A| I  , where I is the identity matrix of order n.

2. For a zero matrix 0, we have adj(0) = 0.

3. For an identity matrix I, we have adj(I)=I.

4. For any scalar K,
   adj (k A) = k^n-1 adj (A)

5. adj (A^T) = (adj A)^T

6. The determinant of the adjoint is: |adj A| =(|A|)^n-1

7. If A is invertible and A^-1 is its inverse, then:

   • adj A = (|A|)A^-1

   • adj A is invertible with inverse (|A|)^-1A

   • adj (A^-1) = (adj A)^-1

8. If A and B are square matrices of the same order, then: adj(AB)=(adjB)(adjA)

9. For any positive integer p, adj (A^p) = (adj A)^p

10. If A is invertible, the same property also works for negative integers P.

Practice Questions

1. Find the adjoint of 
             [ 3   5 ]
      A =  [ 2   7 ]   
2. Find the adjoint of the 3×3 matrix
           [ 1   2    3]
     A = [  0   4   5]  
            [ 1   0    6] 
   

3. A company uses a 2×2 matrix to model a set of product inputs and outputs.

   Matrix A represents: 

             [ 4   1 ]
      A =  [ 3   2 ]

4. A system of 3 linear equations is represented by a matrix                                  

          [  2  -1   3]
     A =[  1   0   4]  
          [   0  5   -2]

   Find adj ( 𝐴 ) and check if the inverse exists.

5. The matrix

             [ 1   2 ]
      A =  [ 3   6 ] 
   represents the transformation of points on a plane. Check whether this matrix can be inverted using the adjoint method.
   
   

FAQs on Adjoint of a Matrix

1. What is the formula for the adjoint of a matrix?

Ans: The adjoint of a square matrix A = [ a_ij ]_n*n is defined as the transpose of the matrix [ a_ij ]_n*n, where Aij is the cofactor of the element a_ij.

2. What is the adjoint of a real matrix?

Ans: The adjoint of a matrix B can be defined as the product of B with its adjoint, yielding a diagonal matrix whose diagonal entries are the determinant det(B). B adj(B) = adj(B) B = det(B) I, where I is an identity matrix. Suppose C is another square matrix then, adj(BC) = adj(C) adj(B).

3. How do you solve a matrix by the adjoint method?

Ans: The adjoint method formulates the gradient of a function towards its parameters in a constrained optimization form. By using the dual form of this constraint optimization problem, it can be used to calculate the gradient very fast.

4. What is the adj matrix?

Ans: In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or adjoint , though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.

5. How to find the adjoint of a 4x4 matrix?

Ans: The adjoint of a matrix is the transpose of the matrix of its cofactors. First, we determine the cofactor of each element of the matrix. Then we form the cofactor matrix using these. Finally, we take the transpose of the cofactor matrix to obtain the adjoint.