Application of Minors and Cofactors

Minors and Cofactors are used in the calculation of the following terms:

• Adjoint of Matrix
• Inverse of Matrix

Adjoint of Matrix

To calculate the adjoint of Matrix, you need to follow the following steps:

Step 1: Calculate the cofactors of each element of a given matrix.

Step 2: Construct the matrix from the cofactor of elements.

Step 3: Calculate the Transpose of the resultant matrix in Step 2.

Step 4: Resulting matrix of Step 3 is the adjoint of the given matrix.

Example: Find the adjoint of the following matrix A;

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

Solution:

Step 1: Compute the cofactors of each element in A.

C₁₁ = 5 × 9 – 6 × 8 = -3
C₁₂ = -(4 × 9 – 6 × 7) = 6
C₁₃ = 4 × 8 – 5 × 7 = -3
C₂₁ = -(2 × 9 – 3 × 8) = 6
C₂₂ = 1 × 9 – 3 × 7 = -6
C₂₃ = -(1 × 8 – 2 × 7) = 3
C₃₁ = 2 × 6 – 3 × 5 = -3
C₃₂ = -(1 × 6 – 3 × 4) = 6
C₃₃ = 1 × 5 – 2 × 4 = -3

Step 2: Construct the matrix of cofactors.

Matrix of cofactors, [Tex]C = \begin{bmatrix} -3&6&-3\\ 6&-6&3\\ -3&6&-3 \end{bmatrix} [/Tex]

Step 3: Transpose the matrix of cofactors.

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

Step 4: The resulting matrix is the adjoint of A.

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

Inverse of Matrix

To calculate the inverse of a matrix, you can use the following steps:

Step 1: Calculate the determinant of the given matrix.

Step 2: If the value of the determinant is zero, then the matrix has no inverse. Otherwise, calculate the adjoint of the matrix using the steps mentioned in the previous heading.

Step 3: Use the formula [Tex]A^{-1} = \frac{adj(A)}{|A|} [/Tex], to calculate the inverse of the given matrix.

Example: Find the inverse of the following matrix A:

[Tex]\bold{A = \begin{bmatrix} 2&-1&0\\ -1 &2 &-1\\ 0&-1&2 \end{bmatrix}} [/Tex]

Solution:

Step 1: Find the determinant of A.

det(A) = 2(2 × 2 – (-1) × (-1)) – (-1)((-1) ×2 – (-1) ×0) + 0 

⇒ det(A) = 6 – 2 = 4

Step 2: Calculate the adjoint of A using the steps mentioned in the previous example.

C₁₁ = 4 – 1 = 3
C₁₂ = -(-2 – 0) = 2
C₁₃ = 1 – 0 = 1
C₂₁ = -(-2 – 0) = 2
C₂₂ = 4 – 0 = 4
C₂₃ = -(-2 – 0) = 2
C₃₁ = (1 – 0) = 1
C₃₂ = -(-2 – 0) = 2
C₃₃ = 4 – 1 = 3

Matrix of cofactors,

[Tex]C = \begin{bmatrix} 3&2&1\\ 2&4&2\\ 1&2&3 \end{bmatrix} [/Tex]

Transpose of matrix of cofactors,

[Tex]adj(A) = C’ =  \begin{bmatrix} 3&2&1\\ 2&4&2\\ 1&2&3 \end{bmatrix} [/Tex]

Step 3: Multiply the adjoint of A by the reciprocal of the determinant.

[Tex] A^{-1} = \frac{adj(A)}{4} = \frac{1}{4} \times\begin{bmatrix} 3&2&1\\ 2&4&2\\ 1&2&3 \end{bmatrix} [/Tex]

[Tex]\Rightarrow A^{-1}= \begin{bmatrix} \frac{3}{4}&\frac{2}{4}&\frac{1}{4}\\ \frac{2}{4}&\frac{4}{4}&\frac{2}{4}\\ \frac{1}{4}&\frac{2}{4}&\frac{3}{4} \end{bmatrix} [/Tex]

[Tex]\Rightarrow A^{-1}= \begin{bmatrix} \frac{3}{4}&\frac{1}{2}&\frac{1}{4}\\ \frac{1}{2}&1&\frac{1}{2}\\ \frac{1}{4}&\frac{1}{2}&\frac{3}{4} \end{bmatrix} [/Tex]

People Also Read:

• Elementary Operation on Matrices
• Order of Matrix

Minors and Cofactors are important to calculate the adjoint and inverse of a matrix. As the name suggests, a Minor is a smaller part of the larger matrix obtained for a particular element of the matrix by deleting the terms of the row and column to which the element belongs. A cofactor is (-1)^i+j times the minor of the matrix.

They are the backbones of Linear Algebra and are used to find the value of a matrix’s determinant, adjoint, and inverse. Other than that there are many use cases in computer science for minors and cofactors. In this article, we will study minors and cofactors in detail. Other than that, we will also learn about the determinants, matrix inversion, and many more.