Adjoint of a Matrix

The adjoint (or adjugate) of a matrix is the transpose of its cofactor matrix.

• It is also called the adjugate matrix.
• The adjoint of a square matrix A is denoted by adj(A)
• Formula: adj(A) = [Cofactor Matrix of A]^T

For any square matrix A, to calculate its adjoint matrix, we have to first calculate the cofactor matrix of the given matrix and then find its determinant.

Steps to Calculate

To calculate the adjoint of a matrix, follow the following steps:

Step 1: Calculate the Minor of all the elements of the given matrix A.
Step 2: Find the Cofactor matrix C using the minor elements.
Step 3: Find the Adjoint matrix of A by taking the transpose of the cofactor matrix C.

Example: Find the Cofactor Matrix of\bold{A =\begin{bmatrix} 1 & 2 & 3\\ 7 & 4 & 5 \\ 6 & 8 & 9 \end{bmatrix}}.

Solution:

Given matrix is A =\begin{bmatrix} 1 & 2 & 3\\ 7 & 4 & 5 \\ 6 & 8 & 9 \end{bmatrix}

Let's find the cofactor of element in first row third column i.e., 3.

Step 1: Delete the entire row and column that contains element under consideration. 
i.e., \begin{bmatrix} \sout{1} & \sout{2} & \sout{3}\\ 7 & 4 & \sout{5} \\ 6 & 8 & \sout{9} \end{bmatrix}

Step 2: Take the remaining elements as it is in the matrix after Step 1.
i.e., \begin{bmatrix} 7 & 4 \\ 6 & 8 \end{bmatrix}

Step 3: Find the determinant of the matrix formed in Step 2 which is called the minor of the element.
Minor of 3 in A  = \begin{vmatrix} 7 & 4 \\ 6 & 8 \end{vmatrix} = 56 - 24 = 32

Step 4: Now use the formula for the cofactor of element a_ij i.e., (-1)^i+j M_ij 
Cofactor of element 3 = (-1)¹⁺³(32) = 32

Step 5: Continue the procedure for all the elements to find the cofactor matrix of A,
i.e., Cofactor Matrix of A =  \begin{bmatrix} -4&-33&32\\ 6&-9&4\\-2&16&-10 \end{bmatrix}

Method to Calculate the Adjoint of a 2×2 Matrix

Let's consider an example for understanding the method to find the adjoint of a 2×2 Matrix.

Example: Find the Adjoint of\bold{\text{A} =\begin{bmatrix}2&3\\ 4&5 \end{bmatrix}}.

Solution:

Given matrix is \text{A} =\begin{bmatrix}2&3\\ 4&5 \end{bmatrix}

Step 1: Find the Cofactor of each element.
Cofactor of element at A[1, 1]: 5
Cofactor of element at A[1, 2]: -4
Cofactor of element at A[2, 1]: -3
Cofactor of element at A[2, 2]: 2

Step 2: Create matrix from Cofactors
i.e.,\bold{\begin{bmatrix}5&-4\\ -3&2 \end{bmatrix}}

Step 3: Transpose of Cofactor matrix,
\bold{Adj(A) =  \begin{bmatrix}5&-3\\ -4&2 \end{bmatrix}}

Method to Calculate

Let's take an example of a 3×3 Matrix :

Example: Find the Adjoint of\bold{A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix}}.

Solution:

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

Step 1: Find the Cofactor of each element.

C₁₁ = (-1)¹⁺¹\begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} = (45 - 48) = -3
C₁₂ = (-1)¹⁺²\begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} = -(36 - 42) = 6
C₁₃ = (-1)¹⁺³\begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} = (32 - 35) = -3
C₂₁ = (-1)²⁺¹\begin{vmatrix} 2 & 3 \\ 8 & 9 \end{vmatrix} = -(18 - 24) = 6
C₂₂= (-1)²⁺²\begin{vmatrix} 1 & 3 \\ 7 & 9 \end{vmatrix} = (9 - 21) = -12
C₂₃ = (-1)²⁺³\begin{vmatrix} 1 & 2 \\ 7 & 8 \end{vmatrix} = -(8 - 14) = 6
C₃₁= (-1)³⁺¹\begin{vmatrix} 2 & 3 \\ 5 & 6 \end{vmatrix} = (12 - 15) = -3
C₃₂ = (-1)³⁺²\begin{vmatrix} 1 & 3 \\ 4 & 6 \end{vmatrix} = -(6 - 12) = 6
C₃₃ = (-1)³⁺³\begin{vmatrix} 1 & 2 \\ 4 & 5 \end{vmatrix} = (5 - 8) = -3

Step 2: Create matrix from Cofactors
C = \begin{bmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \\ \end{bmatrix}

Step 3: Transpose of Matrix C to adjoint of given matrix, rows become columns and columns become row.
\operatorname{adj}(A) = C^{T}= \begin{bmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \\ \end{bmatrix}

Which is adjoint of given matrix A.

Finding the Inverse

To find the inverse of a Matrix using the Adjoint, we can use the following steps:

Step 1: Find the determinant of the matrix.
Step 2: If the determinant is zero, then the matrix is not invertible, and there is no inverse.
Step 3: If the determinant is non-zero, then find the adjoint of the matrix.
Step 4: Divide the adjoint of the matrix by the determinant of a matrix.
Step 5: The result of Step 4 is the Inverse of the given Matrix.

Example: Find the inverse of \bold{A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix}}.

Solution:

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

|A| = 1(45 - 48) -2(36 - 42) + 3(32 - 35)
⇒ |A| = -3 -2(-6) + 3(-3)
⇒ |A| = -3 + 12 - 9 = 0

Thus, inverse of A doesn't exist.

Properties

Adjoints of a matrix have various properties are as follows:

• A(Adj A) = (Adj A)A = |A| I_n
• Adj(BA) = (Adj B) (Adj A)
• |Adj A| = |A|^n-1
• Adj(kA) = k^n-1(Adj A)

Practice Questions on Adjoint of a Matrix