Subtraction of Matrices

A matrix is a rectangular or square array of numbers arranged in rows and columns. If the matrix has m columns and n rows, then it is called m × n. A matrix can be represented as A =  [aij]m×n, where aij denotes the element located in the ith row and jth column, and the values of aij are known as the elements of the matrix.
 

The subtraction of matrices is an operation where the corresponding elements of two matrices are subtracted from one another to create a new matrix.  This operation is similar to matrix addition, and it is only possible when both matrices have the same dimensions.
 

The subtraction of n × n matrices refers to subtracting two corresponding elements of two squares of the same size.  Here, n represents the number of rows and columns in the matrix operation, which indicates the matrix is of order n ×n. For example, there are two matrices, A and B, both of order n × n.  Then, their subtraction A-B is done by subtracting each element of matrix B from the corresponding element in matrix A.

A = [aij]
B = [bij]
Subtraction of matrices = A - B
A - B       = [aij] - [bij]
 

The subtraction of 2 × 2 matrices involves subtracting the corresponding elements of two matrices, each of which has two rows and two columns in the matrix. For example, A and B are two matrices of order 2 × 2; then the difference A-B is calculated by subtracting each element of B from the corresponding element in the A matrix 

A = a2a1  a4a3  

B = b2b1  b4b3  

A - B = a2a1 -b2-b1   a4a3  -b4-b3
 

The Subtraction of 3 × 3 matrices involves subtracting the corresponding elements of two matrices, each having three rows and three columns. This operation only works when two matrices are of the same order (3 × 3). For example, if A and B are the order of matrices in 3 × 3, the difference A-B is calculated by subtracting each element of matrix B from the corresponding element of matrix A.

A = a2a1  a4a3  a6a5

B = b2b1  b4b3 b6b5 

A - B = a2a1 -b2-b1   a4a3  -b4-b3  a6a5 -b6-b5  
 

The subtraction matrix follows some basic rules, similar to the addition matrix, but it does not have all the same properties. Both operations require that the matrices have the same order (same number of rows and columns). However, unlike addition, matrix subtraction does not follow certain laws.

• Matrix subtraction is only defined when both matrices have the same number of rows and columns. This means the number of rows and columns should be of the same order.
• The subtraction of matrices is not commutative, that means A - B  B - A
• The subtraction matrix is not associative, which means (A - B) - C  A (B - C)
• Subtraction of matrices, which subtracts itself, the result is a zero matrix A - A = 0
• Subtraction of matrices  can be written as addition by using the negative of a matrix to another matrix, that is, 
• A - B = A + (-B).
   

The element-wise subtraction of matrices means subtracting each element of one matrix from the corresponding elements in another matrix. This operation is performed position by position, which means subtraction will happen in the same rows and columns. Let's see the example, we have two matrices, A and B:
A = [aij]
B = [bij]
Both have the same size, which is represented by m × n
A-B = [aij - bij]

A= 76 98

B = 32 54

A - B =   7 - 36 - 2 9  - 58 - 4

A - B =  44 44

First row, first column: 6 − 2 = 4

First row, second column: 8 − 4 = 4

Second row, first column: 7 − 3 = 4

Second row, second column: 9 − 5 = 4
 

Subtraction of matrices is not only used to solve math problems, but is also helpful for day-to-day situations. It is used to compare two sets of data, such as the amount of money earned and spent, the amount of stock used, or the weather variations over time. Here are some real-life applications given below:

Inventory management: In inventory management, the subtraction of matrices is used to track changes in the stock level between two time periods. It helps determine how much stock has been used or sold over time.

Finance: In finance, matrix subtraction is used for budget analysis. It helps to compare the budgeted amounts with the actual expenditures. By subtracting the actual spending from the budgeted values, finance teams can easily see over- or underspending across different departments or categories.

Image Editing: The images in digital form can be stored in matrices made of pixels. In image editing, the matrices are helpful to identify the changes between images, such as in motion detection, background removal, or image comparison.

Seating Arrangement Analysis: Matrix subtraction can also be used to analyze the seating capacity. By subtracting the number of occupied seats from the total seats (both stored as matrices), schools can determine how many seats are still available in the hall or auditorium.

Building Construction: Engineers use the subtraction of matrices to subtract the planned material used from the actual material used to monitor resource usage. It helps avoid waste and improve cost efficiency.