What is Matrix in Math – Meaning, Definition & Examples

The knowledge of matrix(plural matrices) is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies our work to a great extent when compared with other straightforward methods. The matrices provide a compact way of solving a system of linear equations. Apart from solving equations, matrices are used in electronic spreadsheet programs for personal computers, and also in physical operations such as magnification, rotation, and reflection through a plane.

Let’s understand what is matrix in math, its properties, and its uses with examples.

What is Matrix in Math?

Matrix, the singular form of matrices, is the arrangement of numbers, variables, symbols, or expressions in a rectangular table that contains various numbers of rows and columns. They are rectangular-shaped arrays, for which different operations like addition, multiplication, and transposition are defined. The numbers or entries in the matrix are known as its elements. Horizontal entries of matrices are called rows and vertical entries are known as columns.

Suppose you want to express the information about the number of notebooks and pens for three friends Ravi, Rakesh, and Saurabh, who have the following items in their bags.

Ravi has 12 notebooks and 3 pens

Rakesh has 8 notebooks and 2 pens

Saurabh has 15 notebooks and 4 pens

Now, this could be arranged in the tabular form as

In a matrix form, it can be expressed as

In the above matrix, the rows denote the three students – Ravi, Rakesh, and Saurabh and the columns denote the number of two items – Notebooks and Pens.

It can also be represented as

In the above matrix, the rows denote the number of two items – Notebooks and Pens and the columns denote the three students – Ravi, Rakesh, and Saurabh.

Definition of a Matrix

A matrix is an ordered rectangular array of numbers consisting of $m$ horizontal rows and $n$ vertical columns. The numbers or functions are called the elements or the entries of the matrix. The matrices are denoted by capital letters.

The above matrix is called an $m \times n$ matrix or matrix of size $m \times n$. For entry, $a_{ij}$, $i$ is the row subscript, while $j$ is the column subscript.

A general matrix is sometimes denoted $[ a_{ij}]_{m \times n}$. 

Examples of Matrices

Example 1: Consider the following matrix.

The size of the matrix is $2 \times 3$ and the entries(or elements) of the matrix are $b_{11} = 1$, $b_{12} = -4$, $b_{13} = 3$, $b_{21} = 5$, $b_{22} = 0$, and $b_{23} = 7$.

Example 2: Consider the following matrix.

The size of the matrix is $1 \times 3$ and the entries(or elements) of the matrix are $p_{11} = 1$, $p_{12} = 2$, and $p_{13} = 9$.

Order of a Matrix

A matrix having $m$ rows and $n$ columns is called a matrix of order $m \times n$ or simply $m \times n$ matrix (read as an $m$ by $n$ matrix). 

So referring to the above examples of matrices, we have $\text{B}$ as $3 \times 2$ matrix, and $\text{P}$ as $3 \times 3$ matrix.

You can see the number of elements in the matrix $\text{B}$ is $6$ and the number of elements in a matrix $\text{P}$ is $3$. 

In general, the number of elements in a matrix of order $m \times n$ is $m \times n = mn$.

Properties of Matrices

Properties of matrices are helpful in performing numerous operations involving two or more matrices. The algebraic operations of addition, subtraction, multiplication, inverse multiplication of matrices, and involving different types of matrices can be easily performed by the use of properties of matrices. The additive, multiplicative identity, and inverse of matrices are also included in this study of the properties of matrices.

The properties of matrices can be broadly classified into the following five properties.

• Properties of Matrix Addition
• Properties of Scalar Multiplication of Matrix
• Properties of Matrix Multiplication
• Properties of Transpose Matrix
• Properties of Inverse Matrix and other properties.

Properties of Matrix Addition

The addition of matrices satisfy the following properties of matrices.

• Commutative Law: For the given two matrixes, matrix $\text{A}$ and matrix $\text{B}$ of the same order, say $m \times n$, then $\text{A} + \text{B} = \text{B} + \text{A}$.
• Associative law: For any three matrices, $\text{A}$, $\text{B}$, $\text{C}$ of the same order $m \times n$, we have $(\text{A} + \text{B}) + \text{C} = \text{A} + (\text{B} + \text{C})$
• Existence of Additive Identity: Let $\text{A}$ be a matrix of order $m \times n$, and $\text{O}$ be a zero matrix or a null matrix of the same order $m \times n$, then $\text{A} + \text{O} = \text{O} + \text{A} = \text{A}$. In other words, $\text{O}$ is the additive identity for matrix addition.
• Existence of Additive Inverse: Let $\text{A}$ be a matrix of order $m \times n$. and let -$\text{A}$ be another matrix of order $m \times n$ such that $\text{A} + (– \text{A}) = (– \text{A}) + \text{A}= \text{O}$. So the matrix $– \text{A}$ is the additive inverse of $\text{A}$ or the negative of matrix $\text{A}$.

Properties of Scalar Multiplication of Matrix

The properties of scalar multiplication of matrix involve a scalar constant and a matrix. For matrixes $\text{A}$ and $\text{B}$ of order $m \times n$, and $k$ and $l$ as scalars values, the property of scalar multiplication of matrices is as follows.

• The product of a constant with the sum of matrices is equal to the sum of the individual product of the constant and the matrix. $k(\text{A} + \text{B}) = k \text{A} + k \text{B}$
• The product of the sum of the constants with a matrix is equal to the sum of the product of each of the constants with the matrix. $(k + l) \text{A} = k \text{A} + l \text{A}$

Note: Both the matrix $\text{A}$ and $\text{B}$ are of the same order and the constants $k$ and $l$ are any real number values.

Properties of Matrix Multiplication

The following properties of matrix multiplication help in performing numerous operations involving matrix multiplication. The condition for matrix multiplication is the number of columns in the first matrix should be equal to the number of rows in the second matrix. 

• Associative Property: For any three matrices $\text{A}$, $\text{B}$, and $\text{C}$ following the matrix multiplication conditions, we have $(\text{AB}) \text{C} = \text{A}( \text{BC})$. Here both sides of the matrix multiplication are defined.
• Distributive Property: For any three matrices $\text{A}$, $\text{B}$, and $\text{C}$ following the matrix multiplication conditions, we have $\text{A}( \text{B} + \text{C}) = \text{AB} + \text{AC}$.
• Existence of Multiplicative Identity: For a square matrix $\text{A}$, having the order $m \times n$, and an identity matrix $\text{I}$ of the same order we have $\text{AI} = \text{IA} = \text{A}$. Here the product of the identity matrix with the given matrix results in the same matrix.

Properties of Transpose Matrix

The properties of matrices for matrices $\text{A}$ and $\text{B}$ of the same order $m \times n$, and a constant $k$ is defined. The following are some of the important properties of the transpose of a matrix.

• The transpose of a matrix on further taking a transpose for the second time results in the original matrix. $(\text{A}^{‘})^{‘} = \text{A}$
• The transpose of the product of a constant and a matrix is equal to the product of the constant and the transpose of the matrix. $(k \text{A})^{‘} = k \text{A}^{‘}$
• The transpose of the sum of two matrices is equal to the sum of the transpose of the individual matrices. $(\text{A} + \text{B})^{‘} = \text{A}^{‘} + \text{B}^{‘}$
• The transpose of the product of two matrices is equal to the product of the transpose of the second matrix and the transpose of the first matrix. $( \text{AB})^{‘} = \text{B}^{‘} \text{A}^{‘}$

Properties of Inverse Matrix and Other Properties

In addition to the above set of properties of matrices, some of the other important properties have been grouped and presented across the below points.

• For a square matrix with real number entries, $\text{A} + \text{A}^{‘} is a symmetric matrix, and $\text{A} – \text{A}^{‘}$ is a skew-symmetric matrix.
• A square matrix can be expressed as a sum of a symmetric and skew-symmetric matrix. $\text{A} = \frac{1}{2}(\text{A} + \text{A}^{‘}) + \frac{1}{2}( \text{A} – \text{A}^{‘})$.
• The inverse of a matrix if it exists is unique. $\text{AB} = \text{BA} = \text{I}$.
• If matrix $\text{A}$ is the inverse of matrix $\text{B}$, then matrix $\text{B}$ is the inverse of matrix $\text{A}$.
• If $\text{A}$ and $\text{B}$ are invertible matrices of the same order $m \times n$, then $(\text{AB})^{-1} = \text{B}^{-1} \text{A}^{-1}$.

Practice Problems

1. What is the meaning of matrix in math?
2. What is meant by an order of a matrix?
3. What are the different properties of matrices?
4. Form the following matrices
   • $\text{A}$ of order $2 \times 2$, where $a_{11} = -1$, $a_{12} = 5$, $a_{21} = 7$, and $a_{22} = 0$
   • $\text{B}$ of order $2 \times 3$, where $b_{11} = 8$, $b_{12} = 7$, $b_{13} = \frac{2}{3}$, $b_{21} = 9$, $b_{22} = 1$, and $b_{23} = 8$
   • $\text{C}$ of order $3 \times 2$, where $c_{11} = 7$, $c_{12} = 8$, $c_{21} = 0$, $c_{22} = -5$, $c_{31} = 11$, and $c_{32} = 1$

FAQs

Conclusion

A matrix is an ordered rectangular array of numbers consisting of $m$ horizontal rows and $n$ vertical columns. The numbers or functions are called the elements or the entries of the matrix. The matrices provide a compact way of solving a system of linear equations. Apart from solving equations, matrices are used in electronic spreadsheet programs for personal computers, and also in physical operations such as magnification, rotation, and reflection through a plane.