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Matrix subtraction is an operation to find the difference between two matrices. Just like matrix addition, the subtraction operation can only be performed if both matrices involved have the same size or order.
The result of matrix subtraction is a new matrix that also has the same order, where each element is the result of subtracting the corresponding elements from the two initial matrices.
There are two common ways to define matrix subtraction, both of which lead to the same result.
Subtraction as Addition with the Opposite
The subtraction of matrix from matrix can be defined as the addition of matrix with the opposite matrix of (i.e., ).
Matrix is the matrix obtained by multiplying each element of matrix by . So, if , then .
Subtraction of Corresponding Elements
If matrix and matrix both have the order , then the result of the subtraction will also be of order .
Each element of matrix is calculated by subtracting the corresponding element of matrix from the corresponding element of matrix :
This means we subtract the elements that are in the same row and column position.
Both definitions are equivalent and will produce the same difference matrix.
To subtract two matrices, follow these steps:
Suppose we have two matrices, and , as follows:
Both matrices are of order , so they can be subtracted.
Using the method of subtracting corresponding elements:
Using the method of addition with the opposite ():
First, determine :
Then, add to :
Both methods yield the same matrix.
Suppose matrix and matrix .
Matrix is of order , while matrix is of order . Since the orders of these two matrices are different, the subtraction (or ) cannot be performed or is undefined.
Matrix addition is commutative and associative, but matrix subtraction generally has neither property.
Not Commutative: In general, the order of matrix subtraction significantly affects the result. This means is not equal to , except in special cases (e.g., if ).
For example, from matrices and above:
Whereas,
Therefore, .
Not Associative: The grouping in the subtraction of three or more matrices also affects the final result. In general, is not equal to .
Matrix subtraction can be written as addition of the opposite matrix: .
By converting subtraction into addition with the opposite matrix, we can use addition properties when needed.
Problem
Given the following matrices:
Determine the result of .
Problem
Determine the values of and from the following matrix equation:
Problem
Given three matrices:
Calculate and . Are the results the same?
Problem
Given:
Then, is:
Problem
Given the matrix equation:
Perform the subtraction operation on the left side:
Based on the equality of two matrices, corresponding elements must be equal:
From the element in row , column :
From the element in row , column :
From the element in row , column :
From the element in row , column : .
Substituting from the first equation gives .
The shared bottom-right value confirms the value of . It adds no new unknown to solve.
The solution is , , and .
Substituting these values into the original equation verifies all four corresponding entries.
Problem
Given:
Calculate :
First, :
Then, :
Calculate :
First, :
Then, :
The results are not the same: .
Matrix subtraction is therefore not associative.
Published: . Updated: .
This is because , whereas .
