Subtraction of Matrices
Subtraction of matrices refers to the subtraction of corresponding elements of two or more matrices. A matrix is a mathematical format for arranging the data in the form of rows and columns. Subtraction of matrices can be done through the elementwise matrix subtraction. Different operations can be applied on matrices such as addition, subtraction and multiplication. In this article, we will mainly focus on the operation of the subtraction of matrices. Subtraction of matrices is a process of subtracting the corresponding elements of the matrices.
Subtraction of matrices is done in the same manner as the addition of matrices. The constraints of matrix addition are applicable for the matrix subtraction as well. Subtraction of matrices is defined only for matrices of the same size. Let us explore the concept in detail using examples.
1.  What is Subtraction of Matrices? 
2.  Subtraction of Matrices of Order 2 × 2 
3.  Subtraction of Matrices of Order 3 × 3 
4.  Properties of Matrix Subtraction 
5.  FAQs on Subtraction of Matrices 
What is Subtraction of Matrices?
Subtraction of matrices is an operation of elementwise subtraction of matrices of the same order, that is, matrices that have the same number of rows and columns. If the number of horizontal rows in a matrix is 'm' and the number of vertical columns is 'n', then the matrix is said to have the dimension 'm × n'. For subtraction of matrices, it is necessary to have the matrices to be subtracted of the same dimension as we subtract the corresponding elements of the matrices.
Subtraction of Matrices Meaning
The subtraction of matrices or matrix subtraction can only be possible if the number of rows and columns of both the matrices are the same. While subtracting two matrices, we subtract the elements in each row and column from the corresponding elements in the row and column of the other matrix. Consider two matrices A and B of the same order 'm × n', where m is the number of rows and n is the number of columns of the two matrices, denoted as A = [a_{ij}] and B = [b_{ij}]. Now, the difference of the two matrices A and B is given as: A  B = [a_{ij}]  [b_{ij}] = [a_{ij } b_{ij}], where ij denotes the position of each element in i^{th} row and j^{th} column. The dimension of the difference matrix, that is, A  B is also 'm × n'.
Subtraction of Matrices of Order 2 × 2
As we know that subtraction of matrices is possible only if the matrices have an equal number of rows and columns, therefore, for the subtraction of matrices of order 2 × 2, the matrices must have 2 rows and 2 columns. Now, consider two matrices A and B with dimensions 2 × 2. To subtract B from A, we will subtract the elements of B from the corresponding elements of A. The general form of subtraction of B from A (order 2 × 2)is:
To understand the concept of matrix subtraction of dimension 2 × 2 better, let us consider an example of two matrices A and B, and subtract B from A.
Subtraction of Matrices of Order 3 × 3
Matrix subtraction of 3 × 3 matrices implies that the matrices to be subtracted from one another have 3 rows and 3 columns. While subtracting matrices, we subtract the elements of one matrix from the corresponding elements of another matrix. The general form of subtraction of matrices A and B of order 3 × 3 is:
Please note that for the subtraction of matrices, matrices need not be square matrices. Matrix subtraction of rectangular matrices is also defined if the order of the matrices is the same.
Properties of Matrix Subtraction
All constraints for the addition of matrices are applied to the subtraction of matrices as well. But there are certain laws that matrix subtraction does not follow just like the subtraction of numbers. The most important necessity for the subtraction of matrices to hold all these properties is that the matrix subtraction is defined only if the order of the matrices is the same.
 The number of rows and columns should be the same for the matrix subtraction.
 The subtraction of matrices is not commutative, that is, A  B ≠ B  A
 The subtraction of matrices is not associative, that is, (A  B)  C ≠ A  (B  C)
 The subtraction of a matrix from itself results in a null matrix, that is, A  A = O.
 Subtraction of matrices is the addition of the negative of a matrix to another matrix, that is, A  B = A + (B).
Important Notes on Subtraction of Matrices
 Subtraction of matrices is possible only if the matrices have the same dimension.
 The subtraction of matrices is not commutative and associative.
 We subtract the corresponding elements of the matrices for the matrix subtraction.
Related Topics to Matrix Subtraction
Examples of Subtraction of Matrices

Example 1: Determine the element in the first row and third column of the matrix B  A using the subtraction of matrices definition if a_{13} = 14 is an element in A and b_{13} = 3 is an element in B
Solution: To determine the element in the first row and the third column of the matrix B  A, we need to calculate the value of b_{13}  a_{13} using matrix subtraction definition
b_{13}  a_{13} =  3  14 = 17
Answer: The element of the first row and third column of B  A is 17.

Example 2: Write the elements of the matrix C = A  B explicitly if A = [2 5 9] and B = [1 9 12] using matrix subtraction formula.
Solution: Since the dimensions of the matrices A and B are the same, that is, 1 × 3, subtraction of matrices is possible for these two matrices.
C = A  B = [21 59 912] = [1 4 3]
Answer: The elements of C = A  B are c_{11} = 1, c_{12} = 4, c_{13} = 3
FAQs on Subtraction of Matrices
What is the Subtraction of Matrices?
Subtraction of matrices is an operation of elementwise subtraction of matrices of the same order, that is, matrices that have the same number of rows and columns. In subtracting two matrices, we subtract the elements in each row and column from the respective elements in the row and column of the other matrix.
How to Do Subtraction of Matrices?
Consider two matrices A and B of the same order 'm × n', where m is the number of rows and n is the number of columns of the two matrices, denoted as, A = [a_{ij}] and B = [b_{ij}]. Now, the difference of the two matrices A and B is given as: A  B = [a_{ij}]  [b_{ij}] = [a_{ij } b_{ij}], where ij denotes the position of each element in i^{th} row and j^{th} column. The dimension of the difference matrix, that is, A  B is also 'm × n'.
Is Subtraction of Matrices Commutative?
The subtraction of matrices is not commutative, that is, A  B ≠ B  A. Just like the subtraction of numbers, subtraction of matrices also has certain constraints.
What is a Necessary Condition for Subtraction of Matrices?
For the subtraction of matrices, the necessary condition is for them to have the same number of rows and columns. Matrices to be subtracted should have the same dimension.
Can we Subtract Two Matrices of Different Orders?
Matrices of different orders cannot be subtracted as when we subtract any two matrices, we subtract the elements of one matrix from the corresponding elements of another matrix. If the orders are different, then corresponding elements are missing in one of the matrices.
Is the Matrix Subtraction Associative?
The matrix subtraction is not associative, that is, (A  B)  C ≠ A  (B  C). Just like the subtraction of numbers, subtraction of matrices also has certain constraints.
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