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Adjoint of a Matrix

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The knowledge of matrices is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics. From matrices there come Determinants, Now we see one of the properties of the Determinant in this article. In this article, we see how to find the Adjoint of a Matrix. To know about the Adjoint of a Matrix we have to know about the Cofactor of a matrix.

Adjoint of a Matrix Definition

The adjoint of a matrix is the transpose matrix of the cofactor of the given matrix. For any square matrix A to calculate its adj. matrix we have to first calculate the cofactor matrix of the given matrix and then find its determinant. To calculate the Ajoint 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.

For any 2×2 matrix A the image of its Adjoint is shown below,

Adjoint of a Matrix

 

Now let’s learn about the Minor, Cofactor, and Transpose of the matrix.

Minor of a Matrix

The minor of the matrix is the matrix or the element that is calculated by hiding the row and column of the matrix of the element for which the minor is calculated. For the 2×2 matrix, the minor is the element that is shown by hiding the row and column of the element for which the Minor is calculated.

Learn more about, Minors and Cofactors 

Cofactor of a Matrix

The Cofactor is the number we get when we remove the column and row of a designated element in a matrix. It means to take one element from a matrix and delete the entire row and column of that element from the matrix, then which elements are present in that matrix, that is called the cofactor. 

How to Find Cofactor of a Matrix

To find the cofactor of an element of a matrix, we can use the following steps:

Step 1: Delete the entire row and column that contains element under consideration. 

Step 2: Take the remaining elements as it is in the matrix after Step 1.

Step 3: Find the determinant of the matrix formed in Step 2 which is called the minor of the element.

Step 4: Now use the formula for the cofactor of element aij i.e., (-1)i+j Mij where Mij is the minor of the element in the ith row and jth column which is already calculated in Step 3.

Step 5: Result of Step 4 is the cofactor of the element under consideration, and similarly, we can calculate the cofactor of each element of the matrix to find the cofactor matrix of the given matrix.

Example: Find 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} 4 & 5 \\ 7 & 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} 4 & 5 \\ 7 & 8 \end{vmatrix} = 32 - 35 = -3

Step 4: Now use the formula for the cofactor of element aij i.e., (-1)i+j Mij 

Cofactor of element 3 = (-1)1+3(-3) = -3

Step 5: Continue the procedure for all the elements to find the cofactor matrix of A,

i.e., Cofactor Matrix of A =  \begin{bmatrix} -3&6&-3\\ 6&12&6\\-3&6&-3 \end{bmatrix} = -3 \begin{bmatrix} 1&2&1\\ -2&-4&-2\\1&-2&1 \end{bmatrix}

Transpose of Matrix

Tranpose of a matrix is the matrix that is formed by changing the rows and columns of the matrix with each other. The transpose of the matrix A is denoted as AT or A. If the order of the matrix A is m×n, then the order of the transpose matrix is n×m.

Learn more about, Transpose of a Matrix

How to find Adjoint of a Matrix?

To find the Adjoint of a Matrix, first, we have to find the Cofactor of each element, and then find 2 more steps. see below the steps,

Step 1: Find the Cofactor of each element present in the matrix.

Step 2: Create another matrix with the cofactors as its elements.

Step 3: Now find the transpose of the matrix which comes from after Step 2.

How to find Adjoint of a 2×2 Matrix

Let’s consider an example for understanding the method to find the adjoint of the 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}}

How to find Adjoint of a 3×3 Matrix

Let’s take an example of a 3×3 Matrix to understand how to calculate the Adjoint of that 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.

C_{12} = (-1)^{1+2} \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} = - (36 - 42) = 6 \\ C_{13} = (-1)^{1+3} \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} = 3 - 28 = -25 \\ C_{21} = (-1)^{2+1} \begin{vmatrix} 2 & 3 \\ 8 & 9 \end{vmatrix} = - (18 - 24) = 6 \\ C_{22} = (-1)^{2+2} \begin{vmatrix} 1 & 3 \\ 7 & 9 \end{vmatrix} = 9 - 21 = -12 \\ C_{23} = (-1)^{2+3} \begin{vmatrix} 1 & 2 \\ 7 & 8 \end{vmatrix} = - (8 - 14) = 6 \\ C_{31} = (-1)^{3+1} \begin{vmatrix} 2 & 3 \\ 5 & 6 \end{vmatrix} = 12 - 15 = -3 \\ C_{32} = (-1)^{3+2} \begin{vmatrix} 1 & 3 \\ 4 & 6 \end{vmatrix} = - (6 - 12) = 6 \\ C_{33} = (-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 & -25 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \\ \end{bmatrix}

Step 3: Transpose of Matrix C to adjoint of given matrix.

\operatorname{adj}(A) = C^{T}= \begin{bmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -25 & 6 & -3 \\ \end{bmatrix}

Which is adjoint of given matrix A.

Properties of Adjoint of a matrix

Adjoint of a matrix have various properties some of those properties are as follows:

  • A(Adj A) = (Adj A)A = |A| In
  • Adj(BA) = (Adj B) (Adj A)
  • |Adj A| = |A|n-1
  • Adj(kA) = kn-1(Adj A)

Finding Inverse Using Adjoint of a Matrix

Finding the inverse is one of the important applications of the Adjoint of the Matrix. To find the inverse of a Matrix using 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.

Learn more about, Inverse of a Matrix

Solved Examples of Adjoint of a Matrix

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

Solution:

Step 1: To find the cofactor of each element

To find the cofactor of each element, we have to delete the row and column of each element one by one and take the present elements after deleting.

Cofactor of elements at A[0,0] = 1 : +\begin{bmatrix} 4 & 5 \\ 8 & 9 \end{bmatrix}          = +(4×9 – 8×5) = -4

Cofactor of elements at A[0,1] = 2 : -\begin{bmatrix} 7 & 5 \\ 6 & 9 \end{bmatrix}          = -(7×9 – 6×5) = -33

Cofactor of elements at A[0,2] = 3 : +\begin{bmatrix} 7 & 4 \\ 6 & 8 \end{bmatrix}          = +(7×8 – 6×4) = 32

Cofactor of elements at A[2,0] = 7 : -\begin{bmatrix} 2 & 3 \\ 8 & 9 \end{bmatrix}          = -(2×9 – 8×3) = 6

Cofactor of elements at A[2,1] = 4 : +\begin{bmatrix} 1 & 3 \\ 6 & 9 \end{bmatrix}          = +(1×9 – 6×3) = -9

Cofactor of elements at A[2,2] = 5 : -\begin{bmatrix} 1 & 2 \\ 6 & 8 \end{bmatrix}          = -(1×8 – 6×2) = 4

Cofactor of elements at A[3,0] = 6 : +\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}          = +(2×5 – 4×3) = -2

Cofactor of elements at A[3,1] = 8 : -\begin{bmatrix} 1 & 3 \\ 7 & 5 \end{bmatrix}          = -(1×5 – 7×3) = 16

Cofactor of elements at A[3,2] = 9 : +\begin{bmatrix} 1 & 2 \\ 7 & 4 \end{bmatrix}          = +(1×4 – 7×2) = -10

The matrix looks like with the cofactors:

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

The final cofactor matrix:

A =\begin{bmatrix} -4 & -33 & 32\\ 6 & -9 & 4 \\ -2 & 16 & -10 \end{bmatrix}

Step 2: Find the transpose of the matrix obtained in step 1

adj(A) =\begin{bmatrix} -4 & 6 & -2\\ -33 & -9 & 16 \\ 32 & 4 & -10 \end{bmatrix}

This is the Adjoint of the matrix.

Example 2: Find the Adjoint of the given matrix A =\begin{bmatrix} -1 & -2 & -2\\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{bmatrix}           .

Solution:

Step 1: To find the cofactor of each element

To find the cofactor of each element, we have to delete the row and column of each element one by one and take the present elements after deleting.

Cofactor of element at A[0,0] = -1 :  +\begin{bmatrix} 1 & -2 \\ -2 & 1 \end{bmatrix}          = +(1×1 – (-2)x(-2)) = -3

Cofactor of elements at A[0,1] = -2 : -\begin{bmatrix} 2 & -2 \\ 2 & 1 \end{bmatrix}          = -(2×1 – 2x(-2)) = -6

Cofactor of elements at A[0,2] = -2 : +\begin{bmatrix} 2 & 1 \\ 2 & -2 \end{bmatrix}          = +(2x(-2) – 2×1) = -6

Cofactor of elements at A[2,0] = 2 : -\begin{bmatrix} -2 & -2 \\ -2 & 1 \end{bmatrix}          = -((-2)x1 – (-2)x(-2)) = 6

Cofactor of elements at A[2,1] = 1 :  +\begin{bmatrix} -1 & -2 \\ 2 & 1 \end{bmatrix}           = +((-1)x1 – 2x(-2)) = 3

Cofactor of elements at A[2,2] = -2 :  -\begin{bmatrix} -1 & -2 \\ 2 & -2 \end{bmatrix}           = -((-1)x(-2) – 2x(-2)) = -6

Cofactor of elements at A[3,0] = 2 : +\begin{bmatrix} -2 & -2 \\ 1 & -2 \end{bmatrix}          = +((-2)x(-2) – 1x(-2)) = 6

Cofactor of elements at A[3,1] = -2 : -\begin{bmatrix} -1 & -2 \\ 2 & -2 \end{bmatrix}           = -((-1)x(-2) – 2x(-2)) = -6

Cofactor of elements at A[3,2] = 1 : +\begin{bmatrix} -1 & -2 \\ 2 & 1 \end{bmatrix}          = +((-1)x(-1)- 2x(-2)) = 3

The final cofactor matrix:

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

Step 2: Find the transpose of the matrix obtained in Step 1

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

This is the Adjoint of the matrix.

FAQs on Adjoint of a Matrix

Q1: What is Adjoint of a Matrix?

Answer:

The adjoint of a square matrix is the transpose of the matrix of cofactors of the original matrix. It is also known as the adjugate matrix.

Q2: How is Adjoint of a Matrix Calculated?

Answer:

To calculate the adjoint of a matrix, you need to find the cofactor matrix of the given matrix and then transpose it.

Q3: What is Use of Adjoint of a Matrix?

Answer:

The key application or use of the adjoint of a matrix is to find the inverse of invertible matrices.

Q4: What is the Relationship between Inverse of a Matrix and its Adjoint?

Answer:

The inverse of a matrix is obtained by dividing its adjoint by its determinant. That is, if A is a square matrix and det(A) is non-zero, then 

A-1 = adj(A)/det(A)

Q5: What is Adjugate Matrix?

Answer:

The adjoint matrix is also called the Adjugate Matrix. It is the transpose of the cofactor of the given matrix.

Q5: What is the Difference between Adjoint and Transpose of a Matrix?

Answer:

The adjoint of a matrix is the transpose of the matrix of cofactors, while the transpose of a matrix is obtained by interchanging its rows and columns. 

Q6: Is a Square Matrix always Invertible?

Answer:

No, square matrix are not always invertible. A square matrix is only invertible if it has a non zero determinant. 

Q7: Can the Adjoint of a Non-Square Matrix be Calculated?

Answer:

No, the adjoint of a matrix can only be calculated for a square matrix due to the definition of it.


Last Updated : 10 Jun, 2023
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