Minor of Matrix

The minor of an element in a matrix is the determinant of the submatrix obtained after the removal of its row and column. For an element a_ij, its minor is denoted by M_ij. Let’s consider a matrix B:
 

\(B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\[0.3em] b_{21} & b_{22} & b_{23} \\[0.3em] b_{31} & b_{32} & b_{33} \end{bmatrix}\)

 

B₁₂ is in row 1, column 2, so we will remove row 1 and column 2 from the given matrix.

The minor of b₁₂ is: 
 

\(M_{12} = \begin{bmatrix} b_{21} & & b_{23} \\ b_{31} & & b_{33} \end{bmatrix}\)
 

Similarly, the minor of each element b_ij in matrix B can be calculated and arranged to form a cofactor matrix M.

\(M = \begin{bmatrix} M_{11} & M_{12} & M_{13} \\[0.3em] M_{21} & M_{22} & M_{23} \\[0.3em] M_{31} & M_{32} & M_{33} \end{bmatrix}\)
 

We should find the minor of an element inside a matrix. To do that, follow the steps given below:

Step 1: Identify the row and column the element belongs to and remove them from the given matrix.
 

Step 2: From the remaining elements, form the submatrix and compute its determinant.    
 

Step 3: Find the determinant of the submatrix to get the minor. Using minor values, form a new matrix called matrix of a minor.
 

For example, consider a 3 × 3 matrix:

\(B = \begin{bmatrix} 1 & 2 & 3 \\[0.3em] 0 & 4 & 5 \\[0.3em] 7 & 6 & 8 \end{bmatrix}\)
 

To find the minor of element b₁₁ remove the first row and first column of the original matrix.

Minor of b₁₁ = M₁₁ = \( \Bigg | \begin{matrix} 4 & 5\\ 6 & 8 \end{matrix} \Bigg|\)

= (4 × 8) – (5 × 6) = 32 – 30 = 2       

The minor of element b₂₃ can be found by removing the second row and third column.  
 
Minor of b₂₃ = M₂₃ = \( \Bigg | \begin{matrix} 1 & 2\\ 7 & 6 \end{matrix} \Bigg|\)

= (1 × 6) – (2 × 7) = 6 – 14 = – 8
                                   

To find the minor of element b₃₂ remove the third row and second column.
 

Minor of b₃₂ = M₃₂ = \( \Bigg | \begin{matrix} 1 & 3\\ 0 & 5 \end{matrix} \Bigg|\)

= (1 × 5) – (3 × 0) = 5 – 0 = 5
 

We can find the minor of each element in this manner. By finding minors of elements, we can form a new matrix that will be the minor of matrix B.

\(M = \begin{bmatrix} M_{11} & M_{12} & M_{13} \\[0.3em] M_{21} & M_{22} & M_{23} \\[0.3em] M_{31} & M_{32} & M_{33} \end{bmatrix} = \begin{bmatrix} 2 & -35 & -28 \\[0.3em] 18 & 9 & -9 \\[0.3em] 11 & 5 & 8 \end{bmatrix}\)

A matrix’s cofactor, determinant, adjoint, and inverse can be determined by finding the minor of a matrix. Let us see the applications of the minor of a matrix.

Cofactor Matrix: C_ij is used to denote the cofactor of an element in a matrix. C_ij is determined by multiplying the minor M_ij and (–1)^{i + j.} 

Therefore, C_ij = (–1)^{i + j}  M_ij

\(\text {Cofactor Matrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\[0.3em] C_{21} & C_{22} & C_{23} \\[0.3em] C_{31} & C_{32} & C_{33} \end{bmatrix}\)

We get cofactor matrix when we replace each element with its cofactor. 

Determinant of a matrix: The determinant of a matrix is a single value that summarizes properties of a matrix. It is calculated using cofactor expansion. The determinant can be found by following these steps:
 

1. Pick one row or column.
2. Find the cofactor of each element in that row/column.
3. Multiply the elements with their cofactors.
4. Add all the products.
    

In a matrix D, the determinant is denoted by |d| or det D. 

The determinant formula for the elements of the first row of matrix D will be:

|d| = d₁₁C₁₁ + d₁₂C₁₂ + d₁₃C₁₃

Here, C_ij = (–1)^i+j  M_ij. So, the determinant of matrix D is:

\(|D| = {{d_{11}(-1)^{1+1} \space \bigg| {\begin{matrix} d_{22} & & d_{23} \\ d_{32} & & b_{33} \end{matrix} } \bigg |} \space + \space {d_{12}(-1)^{1+2}\space \bigg| {\begin{matrix} d_{21} & & d_{23} \\ d_{31} & & b_{33} \end{matrix} } \bigg |} \space + \space {d_{13}(-1)^{1+3} \space \bigg| {\begin{matrix} d_{21} & & d_{21} \\ d_{31} & & b_{32} \end{matrix} } \bigg |}}\)
 

Adjoint of a matrix: To find the adjoint of a 3 × 3 matrix:
 

1. Find the cofactor matrix of the given matrix.
2. Find the transpose of the cofactor matrix.

Let us consider matrix B:
 

\(B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\[0.3em] b_{21} & b_{22} & b_{23} \\[0.3em] b_{31} & b_{32} & b_{33} \end{bmatrix}\)


\(\text {Cofactor (B)} = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\[0.3em] C_{21} & C_{22} & C_{23} \\[0.3em] C_{31} & C_{32} & C_{33} \end{bmatrix}\)


The adjoint of matrix B is equal to the transpose of cofactor matrix

i.e., adj(B) = transpose of cofactor matrix of B


\(adj(B) = [\text {Cofactor (B)}]^T = \begin{bmatrix} C_{11} & C_{21} & C_{31} \\[0.3em] C_{12} & C_{22} & C_{32} \\[0.3em] C_{13} & C_{23} & C_{33} \end{bmatrix}\)
 

Inverse of a matrix: We should divide the adjoint matrix by the determinant to find the inverse of a matrix. 

For a matrix D:

\(D = \begin{bmatrix} d_{11} & d_{12} & d_{13} \\[0.3em] d_{21} & d_{22} & d_{23} \\[0.3em] d_{31} & d_{32} & d_{33} \end{bmatrix}\)
 

The inverse is \(D⁻¹ = {{1 \over det (D)} \times adj(D)}\)


Determinant: |d| = d₁₁C₁₁ + d₁₂C₁₂ + d₁₃C₁₃

Where, C_ij is the cofactor of element d_ij .

Adjoint:

\(adj(D) = \begin{bmatrix} C_{11} & C_{21} & C_{31} \\[0.3em] C_{12} & C_{22} & C_{32} \\[0.3em] C_{13} & C_{23} & C_{33} \end{bmatrix}\)
 

Using inverse formula, If  \(\text {det D} \neq 0\)  the inverse of D is:

\(D⁻¹ = {{1 \over det (D)} \times adj(D)}\)

Knowing how to find minors is essential because they lead to cofactors, determinants, and matrix inverses, which are key concepts in algebra and linear algebra. Here are some tips and tricks for students to master in minor of a matrix. 


 

• Check the base matrix is square: The concept of minors applies to square matrices (n×n). Starting with a non-square matrix can cause confusion.
  
  
   
• Mark the row and column clearly: When finding the minor of element a_ij, draw a line through the i-th row and j-th column and ignore those, it simplifies the step.
  
  
   
• Form the sub-matrix then compute its determinant: After deletion, you get a smaller square matrix. Finding its determinant gives the minor.
  
  
   
• Use consistent notation: Represent the minor of element a_ij as M_ij. This clarity helps when solving determinants or inverses later.
  
  
   
• Link minors to their uses: Remind yourself that minors build up to cofactors and then to the determinant or inverse of the matrix, it gives purpose to mastering the 'minor' step.
      ​

A minor is a building block for the calculation of determinants, cofactors, and matrix inverses. It can be applied to the following real-life applications:
 

• Determining forces and stability in bridges, buildings, or mechanical systems: Engineers use minors and determinants to solve systems of equations that describe the balance of forces.
   

• Predicting the weather using partial differential equations: PDEs are used to simulate the behavior of things like temperature and wind across space and time. These PDEs are converted into linear equations represented using matrices. The minor of a matrix helps find the inverse of the matrix. These matrices are used for error correction and sensitivity analysis of a forecast.
   

• Controlling robotic arms or drones: Robotic control systems require solving linear equations and Jacobian matrices to have precise control of movement. These systems involve minors.
   

• Secure communications using matrix-based encryption: In cryptography, modular inverses require computing minors and cofactors during the encryption and decryption process. 
   

• Applications in complex circuits: Methods like Cramer's rule require minors to compute currents/voltages in complex circuits, like mesh or nodal analysis.