# Matrix Operations – Addition, Subtraction & Multiplication

A matrix in math is an ordered rectangular array of numbers consisting of $m$ horizontal rows and $n$ vertical columns. Similar to ordinary numbers, you can perform basic operations with matrices. The four basic operations that can be performed on matrices are addition, subtraction, scalar multiplication, and matrix multiplication.

Let’s learn about the different types of matrices operations and their properties with examples.

## What Are the Matrix Operations?

The matrix operations help us to combine two or more matrices, to form a single matrix. Similar to numbers, the arithmetic operations of addition, subtraction, and multiplication can also be performed on matrices.

The following are four important matrix operations.

• Subtraction of Matrices
• Scalar Multiplication of Matrices
• Multiplication of Matrices

The addition of matrices is one of the basic operations that is performed on matrices. Two or more matrices of the same order can be added by adding the corresponding elements of the matrices. If $\text{A} = [a_{ij}]$ and $\text{B} = [b_{ij}]$ are two matrices of the same order(with the same dimension), that is, they have the same number of rows and columns, then the addition of matrices $\text{A}$ and $\text{B}$ is $\text{A} + \text{B} = [a_{ij}] + [b_{ij}] = [a_{ij} + b_{ij}]$.

The matrices for addition can be either a square matrix or a rectangular matrix, but the matrices should be of the same order.

#### Properties of Addition of Matrices

The addition of matrices follows similar properties of the addition of numbers: commutative law, associative law, additive inverse, additive identity, etc. The following properties help in the addition matrix operations.

• Commutative Property: Commutative property of matrix addition for the matrices $\text{A} = [a_{ij}]$ and $\text{B} = [b_{ij}]$ of same order $m \times n$, is $\text{A} + \text{B} = \text{B} + \text{A}$.
• Associative Property: Associative property of matrix addition for the matrices $\text{A} = [a_{ij}]$ and $\text{B} = [b_{ij}]$ and $\text{C} = [c_{ij}]$ of same order $m \times n$, is $(\text{A} + \text{B} )+ \text{C} = \text{A} + ( \text{B} + \text{C} )$.
• Additive Identity: Additive identity of matrix addition for a matrix $\text{A} = [a_{ij}]$ of order $m \times n$, is the zero matrix $\text{O}$ of order $m \times n$ such that $\text{A} + \text{O} = \text{O} + \text{A} = \text{A}$.
• Additive Inverse: Additive inverse of matrix addition for the matrix $\text{A} = [a_{ij}]$ of the order $m \times n$, is $-\text{A} = -[a_{ij}]$ of the same order $m \times n$ such that $\text{A} + (-\text{A}) = \text{O} = \text{A} + (-\text{A})$..
• Transpose Property: Transpose property of matrix addition for two matrixes $\text{A} = [a_{ij}]$ and $\text{B} = [b_{ij}]$ of the same order is, $(\text{A} + \text{B})^{\text{T}} = \text{A}^{\text{T}} + \text{B}^{\text{T}}$.
• Determinant Property: Determinant property of matrix addition for two matrixes $\text{A} = [a_{ij}]$ and $\text{B} = [b_{ij}]$ of the same order is, $| \text{A} + \text{B}| = |\text{A}| + |\text{B}|$.

#### Examples of Addition of Matrices

Example 1: Add the matrices $\text{A}$ and $\text{B}$.

The order of matrix $\text{A}$ is $2 \times 2$ and the order of matrix $\text{B}$ is $2 \times 2$, therefore the matrices $\text{A}$ and $\text{B}$ can be added.

Example 2: Add the matrices $\text{P}$ and $\text{Q}$.

The order of matrix $\text{P}$ is $2 \times 2$ and the order of matrix $\text{Q}$ is $2 \times 3$, therefore the matrices $\text{P}$ and $\text{Q}$ cannot be added.

Example 3: Add the matrices $\text{X}$ and $\text{Y}$.

The order of matrix $\text{X}$ is $2 \times 3$ and the order of matrix $\text{Y}$ is $2 \times 3$, therefore the matrices $\text{X}$ and $\text{Y}$ can be added.

### Subtraction of Matrices

Subtraction of matrices is a matrix operation of element-wise subtraction of matrices of the same order, that is, matrices that have the same number of rows and columns. In subtracting two matrices, we subtract the elements in each row and column, from the respective elements in the row and column of another matrix.

Consider two matrices $\text{A}$ and $\text{B}$ of the same order $m \times n$, where $m$ is the number of rows and $n$ is the number of columns of the two matrices, denoted as $\text{A} = [a_{ij}]$ and $\text{B} = [b_{ij}]$. The difference between the two matrices $\text{A}$ and $\text{B}$ is given as $\text{A} – \text{B} = [a_{ij}] – [b_{ij}] = [a_{ij}] −[b_{ij}]$, where $ij$ denotes the position of each element in $i^{th}$ row and $j^{th}$ column. The dimension of the difference matrix, that is, $\text{A} – \text{B}$ is also $m \times n$.

#### Properties of Subtraction of Matrices

The most important necessity for the subtraction of matrices to hold all these properties is that the subtraction of matrices is defined only if the order of the matrices is the same.

• Order of Matrix: The number of rows and columns in the respective matrices should be the same for the subtraction of matrices.
• Commutative Property: The subtraction of matrices is not commutative, that is, $\text{A} – \text{B} \ne \text{B} – \text{A}$.
• Associative Property: The subtraction of matrices is not associative, that is, $(\text{A} – \text{B}) – \text{C} \ne \text{A} – (\text{B} – \text{C})$.
• Null Matrix: The matrix subtraction from itself results in a null matrix, that is, $\text{A} – \text{A} = \text{O}$.
• Negative of a Matrix: Subtraction of matrices is the addition of the negative of a matrix to another matrix, that is, $\text{A} – \text{B} = \text{A} + (-\text{B})$.

#### Examples of Subtraction of Matrices

Example 1: Subtract matrix $\text{A}$ from matrix $\text{B}$.

The order of matrix $\text{A}$ is $2 \times 3$ and the order of matrix $\text{B}$ is $3 \times 2$, therefore the matrices $\text{A}$ and $\text{B}$ cannot be subtracted.

Example 2: Subtract matrix $\text{B}$ from matrix $\text{A}$.

The order of matrix $\text{A}$ is $2 \times 3$ and the order of matrix $\text{B}$ is $2 \times 3$, therefore the matrices $\text{A}$ and $\text{B}$ can be subtracted.

### Scalar Multiplication of a Matrix

The scalar multiplication of a matrix is the product of the scalar constant value with each of the elements of the matrix.

The following properties of scalar multiplication of matrices are helpful in easily performing scalar multiplication of matrices.

#### Properties of Scalar Multiplication of a Matrix

In the following the two matrices are $\text{A}$ and $\text{B}$ and the two scalars are $k$ and $l$.

• $k(\text{A} + \text{B}) = k\text{A} + k\text{B}$
• $(k + l)\text{A} = k \text{A} + l \text{A}$
• $(kl) \text{A} = k(l\text{A}) = l(k \text{A})$

#### Examples of Scalar Multiplication of a Matrix

Example 1: For the give matrix $\text{A}$, find $-2 \text{A}$.

### Multiplication of Matrices

Matrix multiplication is a binary matrix operation performed on matrix $\text{A}$ and matrix $\text{B}$, when both the given matrices are compatible. The primary condition for the multiplication of two matrices is the number of columns in the first matrix should be equal to the number of rows in the second matrix, and hence the order of the matrix is important. The multiplication of matrices does not follow commutative law, $\text{AB} \ne \text{BA}$.

Two matrices $\text{A}$ and $\text{B}$ are said to be compatible if the number of columns in $\text{A}$ is equal to the number of rows in $\text{B}$. The resultant matrix for the multiplication of a matrix $\text{A}$ of order $m \times n$ with a matrix $\text{B}$ of order $n \times p$, is a matrix $\text{C}$ of the order $m \times p$.

For the multiplication of two matrices, the elements of the rows of the matrix are multiplied with the elements of the columns of the next matrix, and the summation of this product results in the elements of the resultant product matrix.

#### Properties of Multiplication of Matrices

The following are the properties of matrix multiplication.

• Matrix Multiplication is Non-Commutative: Matrix multiplication is non-commutative, and the product $\text{AB}$ is not equal to the product $\text{BA}$, i.e., $\text{AB} \ne \text{BA}$.
• Distributive Property: Distributive property over the addition of matrices of matrix multiplication of matrix $\text{A}$ and matrix $\text{B}$ with another matrix $\text{C}$ is $\text{A}(\text{B} + \text{C}) = \text{AB} + \text{BC}$.
• Transpose Property: Transpose property of matrix multiplication for two matrices $\text{A}$ and $\text{B}$ can be given as, $(\text{AB})^{\text{T}} = \text{B}^{\text{T}} \text{A}^{\text{T}}$.
• Complex Conjugate Property: Complex conjugate property of matrix multiplication for two matrices $\text{A}$ and $\text{A}$ is $(\text{AB})^{*} = \text{B}^{*} \text{A}^{*}$.
• Associative Property: Associativity for matrix multiplication for three matrices $\text{A}$, $\text{B}$, and $\text{C}$, such that the products $(\text{AB}) \text{C}$ and $\text{A}( \text{BC})$ are defined, as $(\text{AB}) \text{C} = \text{A}( \text{BC})$.

#### Examples of Multiplication of Matrices

Example 1: Multiply matrix $\text{A}$ by matrix $\text{B}$.

Order of matrix $\text{A}$ is $2 \times 2$ and the order of matrix $\text{B}$ is $2 \times 2$. Since the number of columns of the first matrix $\text{A}$ is equal to the number of rows of the second matrix $\text{B}$, therefore, matrix multiplication $\text{AB}$ is possible.

The order of the resultant matrix $\text{AB}$ will also be $2 \times 2$.

Example 2: For the given matrices $\text{A}$ and $\text{B}$, find

a) $\text{AB}$

b) $\text{BA}$

a) Order of matrix $\text{A}$ is $2 \times 3$ and the order of matrix $\text{B}$ is $2 \times 2$. Since the number of columns of matrix $\text{A}$ is not equal to the number of rows of matrix $\text{B}$($3 \ne 2$), therefore, matrix multiplication $\text{AB}$ is not possible.

b) Order of matrix $\text{B}$ is $2 \times 2$ and the order of matrix $\text{A}$ is $2 \times 3$. Since the number of columns of matrix $\text{B}$ is equal to the number of rows of matrix $\text{A}$($2 = 2$), therefore, matrix multiplication $\text{BA}$ is possible and its order will be $2 \times 3$.

## Practice Problems

For the given matrices, perform the following operations.

• $5 \text{A}$
• $\frac{3}{4} \text{C}$
• $\text{A} + \text{B}$
• $\text{B} + \text{C}$
• $2\text{B} – 3\text{C}$
• $4\text{A} + 2\text{D}$
• $\text{AB}$
• $\text{BA}$
• $\text{BC}$
• $\text{CB}$
• $\text{AD}$
• $\text{DA}$

Which of the above operations is not possible? Give reasons.

## FAQs

### What are the matrix operations?

With matrices, one can perform either of the four basic operations. These operations are matrix addition, matrix subtraction, scalar multiplication of matrix, and matrix multiplication.

### What are the uses of matrix operations?

Matrix operations are commonly used to solve linear equations involving two or more variables. These operations are also used in the field of artificial intelligence and machine learning.

### Why are matrix operations important?

The numbers in a matrix can represent data, and they can also represent mathematical equations. The matrix operations are used in many time-sensitive engineering applications involving many equations with several variables.

### What is the use of matrix in maths?

Matrices are widely used for specifying and representing geometric transformations such as rotations and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions.

## Conclusion

The matrix operations help us to combine two or more matrices, to form a single matrix. The four basic matrix operations are matrix addition, matrix subtraction, scalar multiplication of matrix, and matrix multiplication. Matrix operations are used to solve linear equations involving two or more variables and also in the field of artificial intelligence and machine learning.