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Subtraction of Matrix

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Last updated date: 24th Jul 2024
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An Overview of Subtraction of Matrix

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 \[X = A - B = {a_{ij}} - {b_{ij}}\].


List of Related Articles

  1. Matrices

  2. Operations on matrices

  3. Types of matrices

  4. Matrix addition

FAQs on Subtraction of Matrix

1. What do we get if we subtract an identity matrix of order \[3 \times 3\] from a null matrix of the same order?

We will get the negative of the identity matrix. 

\[\left[ {\begin{array}{*{20}{I}}0&0&0\\0&0&0\\0&0&0\end{array}} \right] - \left[ {\begin{array}{*{20}{I}}1&0&0\\0&1&0\\0&0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{I}}{ - 1}&0&0\\0&{ - 1}&0\\0&0&{ - 1}\end{array}} \right]\]

2. Can we subtract a vector from a matrix?

We can represent a vector in the matrix form. If the order of the given matrix is the same as that of the vector, subtraction is possible in that case.

3. Determine the element in the 2nd row and 5th column of a matrix X – Y, where the element \[{x_{25}} = 17\]and \[{y_{25}} = 20\].

The element in the 2nd row and 5th column of the matrix X – Y is the difference between the elements x25 and y25. Thus, the required number is \[{x_{25}} - {y_{25}} = 17 - 20 =  - 3\].