Subtraction of Matrices is the difference between two matrices which are having the same order. Similar to Addition Matrices, we will do the subtraction for the matrices that are in the same order. We can use the element-wise matrix subtraction to subtract the matrices.
We will take the elements of one matrix and subtract them from the respective elements of the other matrix. Let us know the complete details on the subtraction of the matrices by referring to the entire article. To know about the matrix, addition, subtraction, multiplication, etc information, refer to 10th Grade Math matrix articles on our website.
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Matrix Subtraction – Definition | What is the Subtraction of Matrices?
Subtraction of matrices is the subtraction operation performed on the matrices of the same order. Both subtracting matrices must have the same order which means both must have the same number of rows and columns.
If A = [aij] and B = [bij] are two matrices with the same order or dimension (the same number of rows and columns), then the subtraction of matrices A and B becomes: A – B = [aij] – [bij] = [aij – bij]. ij denotes the position of each element in the ith row and jth column. If matrix A is of 2 × 3 matrices, then the B matrix will also be of 2 × 3 matrices to perform a subtraction operation.
Subtraction of 2 x 2 Matrices | How to Subtract Matrices 2×2?
We can do the subtraction of matrices with the order of 2 × 2 where the matrices are with 2 rows and 2 columns. If you consider two matrices A and B to do subtraction, matrix A must have the order 2 × 2, and matrix B must have the order 2 × 2. The subtraction of two matrices A and B is
Subtraction of 3 x 3 Matrices
The subtraction of matrices with the order of 3 × 3 where the matrices are with 3 rows and 3 columns. If we take two matrices A and B to do subtraction, matrix A must have the order 3 × 3, and matrix B must have the order 3 × 3. The subtraction of two matrices A and B is
Properties on Subtraction of Matrices
Mostly all the properties of the addition of matrices are applied to the subtraction of matrices. But some rules are not applicable to the matrix subtraction. Below are the different properties of subtraction of matrices.
- The number of rows and also the number of columns should be the same for the matrix subtraction.
- The matrix subtraction is not commutative, that is, A – B ≠B – A.
- Also, the subtraction of matrices is not associative, that is, (A – B) – C ≠A – (B – C)
- Subtraction of matrices is also the same as the addition of the negative of a matrix to another matrix, that is, A – B = A + (-B).
- The subtraction of a matrix from itself results in a zero matrix or null matrix, that is, A – A = O.
- If K is a scalar and multiplied to the subtraction A – B, then K(A – B) = KA – KB.
Subtraction of Matrices Examples
Check out the below examples will help you to understand the matrix subtraction. Subtracting matrices is very simple if you solve all the problems given below.
Example 1:
Determine the element in the first row and third column of the matrix B – A using the subtraction of matrices definition if a13 = 16 is an element in A and b13 = -5 is an element in B?
Solution:
To find the element in the first row and the third column of the matrix B – A, we need to do the subtraction. Calculate the value of b13 – a13 using the matrix subtraction
b13 – a13 = – 5 – 16 = -21.
Therefore, the element of the first row and third column of B – A is -21.
Example 2:
Write the elements of the matrix C = A – B explicitly if A = [4 10 18] and B = [2 18 24] using matrix subtraction formula?
Solution:
The given matrices are A = [4 10 18] and B = [2 18 24]. The dimensions of the matrices A and B are the same, that is, 1 × 3. We can perform subtraction of matrices as two matrices have the same order.
Subtract the elements of the first matrix with the respective elements of the second matrix.
A – B = [4-2 10-18 18-24] = [2 -8 -6]
Therefore, the elements of C = A – B are c11 = 2, c12 = -8, c13 = -6.
Example 3:
If A and B are two matrices. Then find subtraction of matrices A and B. \( A =\left[
\begin{matrix}
6&10 \cr
18&16 \cr
\end{matrix}
\right]
\) and \( B =\left[
\begin{matrix}
2&6 \cr
16&18 \cr
\end{matrix}
\right]
\)
Solution:
Given matrices are \( A =\left[
\begin{matrix}
6&10 \cr
18&16 \cr
\end{matrix}
\right]
\) and \( B =\left[
\begin{matrix}
2&6 \cr
16&18 \cr
\end{matrix}
\right]
\)
Now, do the subtraction of matrices A and B.
A – B = \( \left[
\begin{matrix}
6&10 \cr
18&16 \cr
\end{matrix}
\right]
\) – \( \left[
\begin{matrix}
2&6 \cr
16&18 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
6 – 2&10 – 6 \cr
18 – 16&16 – 18 \cr
\end{matrix}
\right]
\) = \( \left[
\begin{matrix}
4&4 \cr
2&-2 \cr
\end{matrix}
\right]
\)
Therefore, the answer is A – B = \( \left[
\begin{matrix}
4&4 \cr
2&-2 \cr
\end{matrix}
\right]
\)
Example 4:
Subtract A – B. \( A = \left[
\begin{matrix}
5&6 \cr
4&9 \cr
\end{matrix}
\right]
\) and \( B = \left[
\begin{matrix}
2&1&4 \cr
7&8&9 \cr
\end{matrix}
\right]
\)
Solution:
Given matrices are \( A = \left[
\begin{matrix}
5&6 \cr
4&9 \cr
\end{matrix}
\right]
\) and \( B = \left[
\begin{matrix}
2&1&4 \cr
7&8&9 \cr
\end{matrix}
\right]
\)
The matrix A has the order of 2 × 2. Matrix B has the order of 2 × 3.
We cannot do subtraction for the given matrices as both have different orders.
See More:
FAQs on Subtracting Matrices
1. What is the subtraction of matrices?
The subtraction of matrices is performing subtraction between two matrices having the same order. If the order of the matrices is different, we cannot perform subtraction for the given matrices.
2. How to subtract two matrices?
If you have two matrices such as A and B, then check their order first. If their order is the same, then subtract the elements of the same positions in the matrices.
If matrix A = aij and matrix B = bij, then the subtraction becomes A – B = aij – bij.
3. How to do subtraction for 2×2 matrices?
We can do the subtraction of the elements of 2×2 matrices in the following order. That is
a11 – b11
a12 – b12
a22 – b22
a21 – b21
4. Can we subtract the 4×4 matrix from the 2×2 matrix?
No, we cannot subtract the 4×4 matrix from the 2×2 matrix as both matrices will not have the same order.
5. Is Subtraction of Matrices Commutative?
No, the Subtraction of Matrices is not Commutative. A – B ≠B – A.
Summary
Subtraction of Matrices is similar to subtraction of numbers. But the matrix subtraction has different rules and different properties. We have explained every concept of subtracting matrices. So, go through the entire article to understand how to do subtraction of matrices. Also, you will get to know what is the rule to do subtraction by reading this article.