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 [Tex]\bold{\text{A} =\begin{bmatrix}2&3\\ 4&5 \end{bmatrix}}         [/Tex].

Solution:

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

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.,[Tex]\bold{\begin{bmatrix}5&-4\\ -3&2 \end{bmatrix}} [/Tex]

Step 3: Transpose of Cofactor matrix,

[Tex]\bold{Adj(A) =  \begin{bmatrix}5&-3\\ -4&2 \end{bmatrix}} [/Tex]

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 [Tex]\bold{A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix}}         [/Tex].

Solution:

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

Step 1: Find the Cofactor of each element.

[Tex]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 \\ [/Tex]

Step 2: Create matrix from Cofactors

[Tex]C = \begin{bmatrix} -3 & 6 & -25 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \\ \end{bmatrix} [/Tex]

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

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

Which is adjoint of given matrix A.

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.