Adjoint of a Matrix

The adjoint of a square matrix is a core concept used in various applications, like solving systems of linear equations and computing matrix inverses.

• The adjoint of a matrix is derived by first building its cofactor matrix and then taking the transpose of that matrix.

• Every element in the cofactor matrix is the cofactor of the corresponding element in the original matrix.

• For a square matrix A, its adjoint is indicated as adj(A).

This concept is fundamental in linear algebra and has applications in solving systems of linear equations and matrix theory.

The adjoint or adjugate of a matrix is the transpose of the matrix of the corresponding cofactors. It is used for calculating the inverse of a matrix. Here, we will be learning the properties of adjoint of a matrix:

1. Identity Relation: For any square matrix A of order n the following holds:
   
   \(A \ adj(A)= adj(A) × A = det(A) × I_n\)
   
   Where I_n​ is the identity matrix of order n.
   
    
2. Determinant of Adjoint: The determinant of the adjoint of A is related to the determinant of A by:
   
   \(det(adj(A)) = det(A) ^{n-1}\)
   
   This property is notable only when A is invertible.
   
    
3. Transpose Property: The adjoint of the transpose of A will be equal to the transpose of the adjoint of A:
   
   \(adj(A^T) = Adj(A)^T\)
   
   This property facilitates the computation of adjoints in matrix operations involving transposition.
   
    
4. Scalar Multiplication: For every kind of square matrix A of order n × n and any scalar k, the adjoint of the matrix kA is given by:
   
   \(adj(kA)=k^{n-1} × adj (A).\)
   
   This property is useful when dealing with scalar multiples of matrices.
   
    
5. Product of matrices: For every two square matrices A and B of the same order, the adjoint of their product will be the product of their adjoints taken in reverse order:
   
   \(adj(AB) = adj(B) . adj(A)\)
   
   This property is important in simplifying expressions involving matrix products.
   
    
6. Adjoint of Adjoint: The adjoint of the adjoint of A is related to A by:
   
   \(adj(adj(A))=det(A) ^{n-2}×A\)
   
   The adjoint of an adjoint property is important in the theoretical anticipation of linear algebra.

These properties are fundamental in matrix theory and solve linear equations, computing determinants, and finding inverses of matrices.

The following chart gives the properties of adjoint of a matrix in a structured table:

 

The adjoint of a square matrix is a new matrix that assists us in finding the inverse of the original matrix. To find the adjoint of a matrix, we follow these steps:
 

1. To calculate the adjoint, we need to find the cofactor matrix for every element in the original matrix and calculate its cofactor.
    
2. After getting the cofactor matrix, we need to take its transpose. It means exchanging rows with columns. As a result, the matrix is the adjoint of the original matrix.

 Hence, for every square matrix A of order n × n, the adjoint of A, indicated as adj(A), is given by:

\(Adj(A) = [Cofactor(A)]^T\)

The minor of an element in a matrix is the determinant of the sub-matrix formed by deleting the row and column that contain that element. The matrix containing the minors of all elements is called the minor of the matrix. The matrix, which gives the minor of an element a_ij​ is denoted as m_ij and is calculated as:

\(m_{ij} = det(A_{ij})\)

Where A_ij ​is the sub-matrix obtained by deleting the i-th row and j-th column of A.

Example

Given matrix:

\(A = \begin{bmatrix}{11} & {12} &{13} \\ {21} & {22} &{23} \\{31} & {32} &{33} \end{bmatrix}\)

The minor of the element a₁₁ is the determinant of the sub-matrix obtained by removing the first row and first column:

 
\(M_{11} = \begin{bmatrix} {22} & {23} \\ {32} &{33}  \end{bmatrix}\)

The cofactor of an element in a matrix is a signed value calculated by multiplying its minor by (−1)^{i + j}, where i and j are the row and column indices of the element. The matrix formed by combining the cofactors of all the elements of a matrix is called its cofactor matrix.

Cofactor Formula

\(C_{ij} = (−1)^{i+j} × M_{ij}\)

Where:

• C_ij Is the cofactor of an element a_ij​
   
• m_ij Is the minor of the element a_ij
   
• i And j are the rows and columns lists of the elements.

The transpose of a matrix is a matrix where the elements of row and column of the original matrix get interchanged. If the matrix A of order m × n, its transpose, denoted as A^T, will have dimensions n × m


How to Find the Transpose

To compute the transpose of a matrix, convert each row of the original matrix into a column in the new matrix.

For example, for a matrix A:

\(A = \begin{bmatrix}a_{11} & a_{12} &a_{13} \\ a_{21} & a_{22} &a_{23} \\a_{31} & a_{32} &a_{33} \end{bmatrix}\)

The transpose A^T is:

\(A = \begin{bmatrix}a_{11} & a_{21} &a_{31} \\ a_{12} & a_{22} &a_{32} \\a_{13} &a_{23} &a_{33} \end{bmatrix}\)

To find the adjoint of a matrix, follow the steps mentioned below:

1. Compute the Minor of each element:

For each component​ a_ij in the matrix, remove the i-th row and j-th column to form a sub-matrix. Compute the determinant of this sub-matrix; this value is the minor m_ij of the element ​a_ij  

For example, a₁₁ = 1

Now we need to remove the first row and first column

\( \begin{bmatrix} {4} & {5} \\ {0} &{6}  \end{bmatrix}\)

The determinant will be
det = (4)(6) − (5)(0) = 24
M₁₁ = 24

2. Determine the Cofactor: The cofactor c_ij ​is given by

\(c_{ij} = (-1)^{i+j}. m _ {ij}\)

This accounts for the sign change based on the position of the element.

For example, a₁₁ = 1
C₁₁ = (−1)1 + 1 × 24 = 24

3. Form the Cofactor matrix: Arrange all the cofactors c_ij into a matrix.

Cofactor matrix = \( \begin{bmatrix} {24} & {-20} &{5}\\ {-5} &{6} &{-1}\\{-4} &{4} &{4} \end{bmatrix}\)


4. Transpose the Cofactor matrix: Take the transpose of the cofactor matrix to obtain the adjoint matrix.

Adj(A) = \( \begin{bmatrix} {24} & {-5} &{-4}\\ {-20} &{6} &{4}\\{-5} &{-1} &{4} \end{bmatrix}\)

Adjoint of a 2 × 2 matrix

The adjoint of a square matrix is gained by first calculating its cofactor matrix and then taking the transpose of that matrix. This adjoint is specifically useful in computing the inverse of the original matrix, provided its determinant is non-zero.

Steps to Find the Adjoint of a 2 × 2 matrix

The matrix given is:

\(A= \begin{bmatrix} {a} & {b} \\{c} &{d}  \end{bmatrix}\)
 

1. Compute the Cofactor matrix: For each element, calculate its cofactor:
    

   c₁₁ = d 
   
   c₁₂ = -c
   
   c₂₁ = -b
   
   c₂₂ = a

   
   So, the cofactor matrix is:
    

   \(C= \begin{bmatrix} {d} & {-c} \\{-b} &{a}  \end{bmatrix}\)
    

2. Transpose the Cofactor matrix:
   
   The adjoint is the transpose of the cofactor matrix:
   
   adj(A) = C^T =  \(C= \begin{bmatrix} {d} & {-b} \\{-c} &{a}  \end{bmatrix}\)
   
   Example
   Given the matrix:
   
   \(C= \begin{bmatrix} {3} & {-6} \\{-4} &{8}  \end{bmatrix}\)
    

• 1. Cofactor matrix:
  
  \(C= \begin{bmatrix} {8} & {4} \\{-6} &{3}  \end{bmatrix}\)

   

•  2. Adjoint:
  
  adj(A) = C^T = \(\begin{bmatrix} {8} & {-6} \\{4} &{3}  \end{bmatrix}\)

Adjoint of 3 × 3 matrices

Every adjoint of a square matrix is the transpose of its cofactor matrix. It is essential for calculating the inverse of a matrix, provided the determinant is non-zero.

Step-by-Step Process

The matrix given is:

\(A = \begin{bmatrix}{a_{11}} & {a_{12} }&{a_{13}} \\ a_{21} & a_{22} &a_{23} \\a_{31} & a_{32} &a_{33} \end{bmatrix}\)

1. Compute the minors of all elements:
Calculate m_ij of all elements by deleting the i-th row and j-th column of A, and finding the determinant of the sub-matrix.

2. Compute the cofactors of all elements:

For each element, a_ij​ calculate its cofactor c_ij using the formula:
\(c_{ij}=(-1)^{i+j} . det(m^{ij})\)

3. Form the Cofactor matrix:
Arrange all the cofactors c_ij​ into a matrix.

4. Transpose the Cofactor matrix:

 The adjoint (adjoint) is the transpose of the cofactor matrix:
 adj(A) = C^T

The following chart gives the steps in summarized form for calculating adjoint of a matrix:

 

Practice Problem:
Given the matrix:

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

1. Compute the minors:

\(M_{11} = 5 \times 9 - 6\times 8 = 45 - 48 = -3\\ M_{12} = 4 \times 9 - 6\times 7 = 36 - 42 = -6\\ M_{13} = 4 \times 8 - 5 \times 7 = 32 - 35 = -3 \\ M_{21} = 2 \times 9 = 3 \times 8 = 18 - 24 = -6\\ M_{22} = 1 \times 9 - 3 \times 7 = 9 - 21 = -12\\ M_{23} = 1 \times 8 - 2\times 7 = 8 - 14 = 6\\ M_{31} = 2 \times 6 - 5 \times 3 = 12 - 15 = -3\\ M_{32} = 1 \times6 - 4 \times 3 = 6 - 12 = -6\\ M_{33} = 1 \times 5 - 4 \times 2 = 5 - 8 = -3\)

2. Calculate the cofactors off all elements:

\(C_{11} -3\\ C_{12} = 6\\ C_{13} = -3 \\ C_{21} = 6\\ C_{22} = -12\\ C_{23} = - 6\\ C_{31} = -3\\ C_{32} = 6\\ C_{33} = -3\)

3. Form the Cofactor matrix:

\(A = \begin{bmatrix}{-3} & {6}&{-3} \\ 6 & -12&-6 \\-3 & 6 &-3\end{bmatrix}\)

4. Transpose the Cofactor matrix:

\(A = \begin{bmatrix}{-3} & {6}&{-3} \\ 6 & -12&6 \\-3 & -6 &-3\end{bmatrix}\)

To make understanding of adjoint matrix easy and simple, here are a few essential tips to solve problems effectively:
 

1. For a 2 × 2 matrix, swap the diagonal elements and change the sign to find the adjoint.
    
2. Memorize the sign patterns for calculating cofactor.
    
3. Adjoint of an identity matrix is itself.
    
4. To find the minor of an element a_ij, find the determinant of the matrix after eliminating i-th row and j-th column.
    
5. Remember to take the transpose of cofactor matrix to find the adjoint.

The adjoint or adjoint of a matrix is a core rule in linear algebra with diverse real-life applications. It plays a crucial role in solving systems of linear equations, computing matrix inverses, and analyzing linear transformations. Here are a few such applications: 

1. Geometric Applications or Engineering Systems:
   
   For analyzing electric circuit, nodes, mesh analysis, we use adjoints for solving linear equations. Similarly, for calculating the area of parallelogram, determinants are used for accurate measurement.
   
    
2. Eigenvalue Problems and Quantum Mechanics:
   
   In quantum mechanics, the adjoint of a matrix is used in the study of operators and their eigenvalues. It provides insights into the properties of linear transformations, which are fundamental in understanding quantum states and observables.
   
    
3. Optimization and Sensitivity Analysis:
   
   Adjoint methods are employed in optimization problems, especially in high-dimensional spaces. For example, using adjoint methods by many engineers can efficiently calculate the gradient of a performance metric, like lift-to-drag ratio, concerning numerous design parameters. 
   
    
4. Computer Graphics and Data Transformations:
   
   In computer graphics, matrices are used to perform transformations such as scaling, rotation, and translation of objects. It enables to perform operations like object inversion and camera transformations.
   
    
5. Cryptography and Coding Theory:
   
   Matrices and their inverses are fundamental in cryptography, particularly in classical encryption methods like the Hill cipher, which is essential for decoding messages. Moreover, in coding theory, matrices are employed to design error-correcting codes, ensuring reliable data transmission.