How to Subtract Matrices Formula Properties and Solved Examples
A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has ‘m’ rows and ‘n’ columns,, then the matrix's order is \[m \times n\]. Such a matrix can mathematically be represented as \[A = {\left[ {{a_{ij}}} \right]_{m \times n}}\], where the numbers written in the matrix, that is, \[{a_{ij}}\] which belongs to ith row and jth column, are known as the elements of the matrix.
There are different types of matrices like symmetric matrix, skew–symmetric matrix, zero matrices, square matrices, identity matrices, etc. We can also perform algebraic operations on the matrices: addition, subtraction and multiplication.
To add, subtract or multiply two matrices, we first need to check if the required condition is satisfied by order of the given matrices. For addition and subtraction, the order of the matrices should be equal, while in multiplication, the number of columns in the first matrix should be equal to the number of rows in the second matrix.
What is Subtraction of Matrices?
Subtraction of two matrices refers to finding the difference between the corresponding elements of the matrices. Subtraction can only be performed on the matrices having the same order. Given two matrices \[A = {a_{ij}}\] and \[B = {b_{ij}}\], the difference of A and B matrix is X, then X can be defined as \[X = A - B = {a_{ij}} - {b_{ij}}\]. This means that the elements of matrix X are the difference between corresponding elements of A and B. The order of the new matrix formed is the same as the initial matrices.
How to Subtract Two Matrices?
To subtract two matrices, first ensure that the given matrices are of the same order. Then we need to perform element–wise subtraction of the corresponding elements of the two given matrices and then arrange the difference in the corresponding positions. Let's demonstrate the above using an example.
Given two matrices A and B of order \[2 \times 3\].
Let \[A = \left[ {\begin{array}{*{20}{I}}{ - 1}&2&6\\4&9&0\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}7&{ - 4}&5\\8&{13}&{ - 2}\end{array}} \right]\]
The difference between the A and B matrices is X, which can be found by performing the following steps.
X
= A - B
\[= \left[ {\begin{array}{*{20}{I}}{ - 1}&2&6\\4&9&0\end{array}} \right] - \left[ {\begin{array}{*{20}{I}}7&{ - 4}&5\\8&{13}&{ - 2}\end{array}} \right]\]
\[= \left[ {\begin{array}{*{20}{I}}{ - 1 - 7}&{2 - \left( { - 4} \right)}&{6 - 5}\\{4 - 8}&{9 - 13}&{0 - \left( { - 2} \right)}\end{array}} \right]\]
\[= \left[ {\begin{array}{*{20}{I}}{ - 8}&6&1\\{ - 4}&{ - 4}&2\end{array}} \right]\]
Properties of Matrix Subtraction
The order of the matrices to be subtracted should be equal.
Matrix subtraction is not commutative, that is, A – B \[ \ne \]B – A.
Matrix subtraction is not associative, that is, A – (B – C) \[ \ne \] (A – B) – C.
Subtraction of matrices can also be interpreted as the addition of the negative of the second matrix to the first matrix, that A – B = A + (-B).
Solved Examples
Q1. If A and B are two matrices. Calculate the subtraction of matrices A and B.
\[A = \left[ {\begin{array}{*{20}{I}}9&2\\5&4\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}{ - 1}&6\\0&{ - 5}\end{array}} \right]\].
Ans. The difference between A and B is
A - B
\[= \left[ {\begin{array}{*{20}{I}}9&2\\5&4\end{array}} \right] - \left[ {\begin{array}{*{20}{I}}{ - 1}&6\\0&{ - 5}\end{array}} \right]\]
\[ = \left[ {\begin{array}{*{20}{I}}{9 - \left( { - 1} \right)}&{2 - 6}\\{5 - 0}&{4 - \left( { - 5} \right)}\end{array}} \right]\]
\[ = \left[ {\begin{array}{*{20}{I}}{10}&{ - 4}\\5&9\end{array}} \right]\]
Q2. Calculate 2A – B, such that,
\[A = \left[ {\begin{array}{*{20}{I}}{ - 1}&9\\0&{ - 4}\\6&3\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}{10}&3\\{15}&{ - 7}\\{ - 11}&{ - 6}\end{array}} \right]\].
Ans. First calculate the value of 2A,
2A
\[= 2\left[ {\begin{array}{*{20}{I}}{ - 1}&9\\0&{ - 4}\\6&3\end{array}} \right]\]
\[= \left[ {\begin{array}{*{20}{I}}{ - 1 \times 2}&{9 \times 2}\\{0 \times 2}&{ - 4 \times 2}\\{6 \times 2}&{3 \times 2}\end{array}} \right]\]
\[ = \left[ {\begin{array}{*{20}{I}}{ - 2}&{18}\\0&{ - 8}\\{12}&6\end{array}} \right]\]
Now subtract B from 2A,
\[2A - B\]
\[ = \left[ {\begin{array}{*{20}{I}}{ - 2}&{18}\\0&{ - 8}\\{12}&6\end{array}} \right] - \left[ {\begin{array}{*{20}{I}}{10}&3\\{15}&{ - 7}\\{ - 11}&{ - 6}\end{array}} \right]\]
\[ = \left[ {\begin{array}{*{20}{I}}{ - 12}&{15}\\{ - 15}&{ - 1}\\{23}&{12}\end{array}} \right]\]
Practice Questions
Q1. Calculate 3A – 2B, such that,
\[A = \left[ {\begin{array}{*{20}{I}}3&{ - 17}\\1&{ - 40}\\8&{21}\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}0&{13}\\{56}&{ - 71}\\{ - 100}&6\end{array}} \right]\].
Ans. \[\left[ {\begin{array}{*{20}{I}}9&{ - 77}\\{ - 109}&{22}\\{224}&{51}\end{array}} \right]\]
Q2. A shopkeeper arranges the prices of the ornaments he purchased in matrix form and the selling prices of the same ornaments. Calculate the profit or loss in each case.
\[CP = \left[ {\begin{array}{*{20}{I}}{400}\\{250}\\{720}\\{825}\end{array}} \right]\] and \[SP = \left[ {\begin{array}{*{20}{I}}{300}\\{500}\\{800}\\{825}\end{array}} \right]\].
Ans. \[\left[ {\begin{array}{*{20}{I}}{100}\\{ - 250}\\{ - 80}\\{0}\end{array}} \right]\]
Q3. Determine the element in the 3rd row and 8th column of a matrix X – Y, where the element \[{x_{38}} = 171\]and \[{y_{38}} = 120\].
Ans. 51
Interesting Facts
The subtraction of a matrix from itself produces a null matrix, that is, A – A = 0.
A scalar quantity k can be distributed on the subtraction of two matrices, that is, k(A – B) = kA – kB.
If a null matrix is subtracted from a given matrix we will get the matrix itself, that is, A – 0 = A.
Key Features
A matrix is a rectangular array of numbers arranged in the form of rows and columns. If a matrix has ‘m’ rows and ‘n’ columns then the order of the matrix is \[m \times n\] and can be mathematically be represented as \[A = {\left[ {{a_{ij}}} \right]_{m \times n}}\].
Matrices can be added, subtracted and multiplied.
For addition and subtraction of matrices, the order of the matrices should be the same.
For the multiplication of matrices, the number of columns of the first matrix should be equal to the number of rows of the second matrix.
Subtraction of two matrices refers to the process of finding the difference between the corresponding elements of the matrices. Given two matrices \[A = {a_{ij}}\] and \[B = {b_{ij}}\], the difference of A and B matrix is X, then X can be defined as \[X = A - B = {a_{ij}} - {b_{ij}}\].
Subtraction of matrices is neither commutative nor associative.
Subtraction of matrices can also be interpreted as the addition of the negative of the second matrix to the first matrix, that A – B = A + (-B).
Conclusion
A matrix is a rectangular array of numbers arranged in the form of rows and columns. If a matrix has ‘m’ rows and ‘n’ columns then the order of the matrix is \[m \times n\] and can be mathematically be represented as \[A = {\left[ {{a_{ij}}} \right]_{m \times n}}\]. For addition and subtraction of matrices, the order of the matrices should be the same.
Subtraction of two matrices refers to the process of finding the difference between the corresponding elements of the matrices. Given two matrices \[A = {a_{ij}}\] and \[B = {b_{ij}}\], the difference of A and B matrix is X, then X can be defined as \[X = A - B = {a_{ij}} - {b_{ij}}\].
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FAQs on Subtraction of Matrices Explained with Steps and Examples
1. What is subtraction of matrices?
The subtraction of matrices is the process of subtracting corresponding elements of two matrices of the same order. If A and B are matrices of order m × n, then their difference is given by:
A − B = [aij − bij].
This means:
- Subtract each element in matrix B from the corresponding element in matrix A.
- Both matrices must have the same number of rows and columns.
2. What is the formula for subtraction of matrices?
The formula for matrix subtraction is (A − B)ij = aij − bij, where aij and bij are corresponding elements of matrices A and B.
In general:
- If A = [aij] and B = [bij]
- Then A − B = [aij − bij]
3. How do you subtract two matrices step by step?
To subtract two matrices, subtract each corresponding element of the second matrix from the first matrix.
Steps:
- Check that both matrices have the same order.
- Subtract elements in the same position.
- Write the results in the same position in the new matrix.
If A = [[5, 3], [2, 1]] and B = [[1, 2], [4, 0]],
A − B = [[5−1, 3−2], [2−4, 1−0]] = [[4, 1], [−2, 1]].
4. Can you subtract matrices of different orders?
No, you cannot subtract matrices of different orders because matrix subtraction requires the same number of rows and columns.
For example:
- A is 2 × 2 and B is 3 × 2 → subtraction is not defined.
5. Is matrix subtraction commutative?
No, matrix subtraction is not commutative, meaning A − B ≠ B − A in general.
Example:
If A = [[3]] and B = [[1]],
A − B = 2 but B − A = −2.
Since 2 ≠ −2, subtraction of matrices does not satisfy the commutative property.
6. What are the properties of subtraction of matrices?
The main properties of matrix subtraction describe how matrices behave under subtraction.
Important properties:
- Not commutative: A − B ≠ B − A
- Not associative: (A − B) − C ≠ A − (B − C)
- A − O = A, where O is the zero matrix
- A − A = O
7. What is the result when a matrix is subtracted from itself?
When a matrix is subtracted from itself, the result is the zero matrix of the same order.
Mathematically:
A − A = O
Each element becomes zero because aij − aij = 0 for all i and j.
8. What is the difference between matrix addition and matrix subtraction?
The difference between matrix addition and matrix subtraction lies in whether corresponding elements are added or subtracted.
- Addition: (A + B)ij = aij + bij
- Subtraction: (A − B)ij = aij − bij
9. How is matrix subtraction related to adding a negative matrix?
Matrix subtraction can be written as addition of the negative matrix: A − B = A + (−B).
Here:
- −B is obtained by multiplying each element of B by −1.
- Then add it to matrix A.
10. Can you give a real-life application of subtraction of matrices?
Subtraction of matrices is used in real life to compare data sets such as profit and loss, inventory changes, or performance differences.
For example:
- Matrix A represents sales in 2024.
- Matrix B represents sales in 2025.
- A − B gives the change in sales for each product and region.
