Solved Examples of Adjoint of a Matrix

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

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 : [Tex]+\begin{bmatrix} 4 & 5 \\ 8 & 9 \end{bmatrix}          [/Tex] = +(4×9 – 8×5) = -4

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

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

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

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

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

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

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

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

The matrix looks like with the cofactors:

[Tex]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} [/Tex]

The final cofactor matrix:

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

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

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

This is the Adjoint of the matrix.

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

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 :  [Tex]+\begin{bmatrix} 1 & -2 \\ -2 & 1 \end{bmatrix}          [/Tex] = +(1×1 – (-2)x(-2)) = -3

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

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

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

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

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

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

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

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

The final cofactor matrix:

[Tex]A =\begin{bmatrix} -3 & -6 & -6\\ 6 & 3 & -6 \\ 6 & -6 & 3 \end{bmatrix} [/Tex]

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

[Tex]adj(A) =\begin{bmatrix} -3 & 6 & 6\\ -6 & 3 & -6 \\ -6 & -6 & 3 \end{bmatrix} [/Tex]

This is the Adjoint of the matrix.

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.