Subtraction of Matrices

Subtraction of matrices involves subtraction of corresponding elements from two or more matrices. However, before we go ahead to discuss a subtraction of matrices, let us understand what a matrix is. A matrix is a mathematical format where data and elements are arranged with the help of rows and columns. 

Generally, subtraction of matrices is done through element-wise matrix subtraction. Different operations can be applied to matrices, including addition, subtraction, and multiplication.

In this article, our primary focus would be to elaborate on the subtraction of matrices. The subtraction of matrices is done quite the same way as the addition of matrices. This means that the conditions applicable for matrix addition and similar conditions are applied for matrix subtraction. 

Subtraction of Matrices

The subtraction of matrices is an operation where element-wise subtraction applies to the matrices of the same order, which essentially means that subtraction between two matrices can only happen when both of them have the same number of rows and columns. 

If the number of horizontal rows in a matrix is ‘m’ and the number of columns is ‘n’, the matrix is of order x n’. For subtraction of matrices, both the matrices must be of the same order to subtract the corresponding elements of the matrices.

So let us take an example. 

There is a matrix A=[aij]m x n and B= [bij] m x n 

So now

If A=     B=  

Then 

A-B=

Subtraction of Matrices of Order 2 x 2

Since we know that subtraction of matrices is only possible when both the matrices have an equal number of columns and rows, for the subtraction of matrices of order 2 x 2, the matrices should have two rows as well 2 columns.

Let us take an example. 

Consider two matrices, A and B, with order 2 x 2. So, to subtract B from A, we need to subtract the elements of B from the corresponding elements of A. 

So, if 

A=   and  B =

Then

A-B =  

Subtraction of Matrices of order 3 x 3 

The same principle will work in the case of 3 x 3 matrices where the matrices are to be subtracted from one another with 3 rows and 3 columns. For subtracting the matrices, we need to subtract the elements from one matrix with the corresponding elements of the other matrix.

So, if 

A=     B=

Then

A-B=

A-B=

Properties of Matrix Subtraction

The conditions for the addition of matrices also applies to the subtraction of matrices. However, there are some exceptions. There are certain laws that the matrix subtraction does not follow. So let us look at some of the properties and conditions for matrix subtraction.

1. The number of rows and columns should be the same for a matrix subtraction operation.

2. The subtraction of matrices is not commutative, which means A-B ≠ B-A.

3. Also, the subtraction of matrices is not associative, which means (A-B) – C ≠ A- (B-C).

4. The subtraction of a matrix from itself results in a null matrix; this means A-A= 0.

5. The subtraction of matrices is the addition of the negative of a matrix to another matrix which means A – B = A + (-B).

In Subtraction of Matrix, we generally use

Element Wise Subtraction of Matrices

In this type, for subtracting two matrices, we subtract the elements in each row and a column to the corresponding elements of the rows and columns of the next matrix.

So for example A= [aij]m x n and B =[bij] m x n, are two matrices so by subtracting them we get A-B= [aij]mxn – [bij] m x n = [aij – bij] mxn where ij denotes the position of each element in ith row and jth column.

Examples of Subtraction of Matrices

• Determine the element in the first row and the third column of the matrix Q-P by using the subtraction of matrices definition where p13 =14 is an element in P and q13 = -3 is an element in B.

Now let us try to solve it.

To determine the element in the first row and the third column of the matrix Q-P, we first need to calculate the value of q13 – p13  by using the matrix subtraction

q13 – p13 = -3 -14 = -17.

So, we can say the element of the first row and the third column of Q-P is -17.

• Write the elements of the matrix C= A-B if A= [2 5 9] and B = [ 1 9 12] by using the subtraction of order of matrices operation

So first, we have to check whether the order of matrices A and B are the same, and yes, they are the same, i.e., they have the order 1 x 3. So, subtraction of the matrices are possible 

Hence it will be 

C= A – B = [ 2-1    5-9    9-12] = [ 1 -4 -3]. So, we can say that c11= 1, c12 = -4 and c13 = -3.

Conclusion

Hopefully, by the examples and the different properties, you have a thorough understanding of the subtraction of matrices and the conditions at which the operation can be applied.