• Home
• /
• Blog
• /
• Types of Matrices(With Properties & Examples)

# Types of Matrices(With Properties & Examples)

February 13, 2023

A matrix in math is an ordered rectangular array of numbers consisting of $m$ horizontal rows and $n$ vertical columns. There are various types of matrices. And these are used in multiple areas such as finding solutions to linear equations, in electronic spreadsheet programs for personal computers, and in physical operations such as magnification, rotation, and reflection through a plane.

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

## Types of Matrices

The matrices are distinguished on the basis of their order, elements, and certain other conditions. There are different types of matrices but the most commonly used are as follows.

• Row Matrix
• Column Matrix
• Horizontal Matrix
• Vertical Matrix
• Rectangular Matrix
• Square Matrix
• Zero or Null Matrix
• Diagonal matrix
• Scalar Matrix
• Unit or Identity Matrix

### Row Matrix

A matrix that has only one row is known as a row matrix. Thus $\text{A} = [a_{ij}]_{m \times n}$ is a row matrix if $m = 1$. In other words, we can say that a matrix $\text{A}$ is called a row matrix, if it’s of order $1 \times n$.

#### Examples of Row Matrix

Example 1: A student has $8$ notebooks, $5$ textbooks, $2$ pens, $3$ pencils, and $1$ eraser in his bag. Represent the number of different items in the form of a row matrix.

The above items can be represented in the form of a row matrix as

Example 2: Pawan has $12$ red marbles, $23$ green marbles, $17$ yellow marbles, and $5$ blue marbles. Represent the number of marbles of different colours in the form of a row matrix.

The number of marbles can be represented in the form of a row matrix as

### Column Matrix

A matrix that has only one column is known as a column matrix. Thus $\text{A} = [a_{ij}]_{m \times n}$ is a column matrix if $n = 1$. In other words, we can say that a matrix $\text{A}$ is called a column matrix, if it’s of order $m \times 1$.

#### Examples of Column Matrix

Let’s consider the above examples to understand what are column matrices.

Example 1: A student has $8$ notebooks, $5$ textbooks, $2$ pens, $3$ pencils, and $1$ eraser in his bag. Represent the number of different items in the form of a column matrix.

The above items can be represented in the form of a column matrix as

Example 2: Pawan has $12$ red marbles, $23$ green marbles, $17$ yellow marbles, and $5$ blue marbles. Represent the number of marbles of different colours in the form of a column matrix.

The number of marbles can be represented in the form of a column matrix as

### Horizontal Matrix

A matrix where the number of columns is greater than the number of rows is called a horizontal matrix.  A matrix $\text{A} = [a_{ij}]_{m \times n}$ is called a horizontal matrix, if $n \gt m$.

#### Examples of Horizontal Matrix

Example 1: There are two students in a class – student1 and student2. Student1 has $5$ notebooks, $2$ textbooks, $1$ pen, $2$ pencils, and $1$ eraser in his bag and student2 has $7$ notebooks, $6$ textbooks, $1$ pen, $4$ pencils, and $2$ eraser in his bag. Represent the number of different items in the form of a horizontal matrix.

The above items can be represented in the form of a horizontal matrix as

Example 2: Three friends, Pawan, Manoj, and Saurabh were counting the number of marbles of different colours they have.

The number of marbles of each colour with three of them is given as Pawan has $14$ red, $19$, green, $21$ yellow, and $15$ blue marbles, Manoj has $18$ red, $7$, green, $12$ yellow, and $6$ blue marbles, and Saurabh has $10$ red, $5$, green, $30$ yellow, and $7$ blue marbles. Represent the number of marbles of different colours in the form of a horizontal matrix.

The number of marbles can be represented in the form of a horizontal matrix as

### Vertical Matrix

A matrix where the number of rows is greater than the number of columns is called a vertical matrix.  A matrix $\text{A} = [a_{ij}]_{m \times n}$ is called a vertical matrix, if $m \gt n$.

#### Examples of Vertical Matrix

Let’s consider the above examples to understand what are vertical matrices.

Example 1: There are two students in a class – student1 and student2. Student1 has $5$ notebooks, $2$ textbooks, $1$ pen, $2$ pencils, and $1$ eraser in his bag and student2 has $7$ notebooks, $6$ textbooks, $1$ pen, $4$ pencils, and $2$ eraser in his bag. Represent the number of different items in the form of a vertical matrix.

The above items can be represented in the form of a vertical matrix as

Example 2: Three friends, Pawan, Manoj, and Saurabh were counting the number of marbles of different colours they have.

The number of marbles of each colour with three of them is given as Pawan has $14$ red, $19$, green, $21$ yellow, and $15$ blue marbles, Manoj has $18$ red, $7$, green, $12$ yellow, and $6$ blue marbles, and Saurabh has $10$ red, $5$, green, $30$ yellow, and $7$ blue marbles. Represent the number of marbles of different colours in the form of a vertical matrix.

The number of marbles can be represented in the form of a vertical matrix as

### Rectangular Matrix

A matrix where the number of rows is not equal to the number of columns is called a rectangular matrix.  A matrix $\text{A} = [a_{ij}]_{m \times n}$ is called a rectangular matrix, if $m \ne n$.

#### Examples of Rectangular Matrix

In the above examples, row matrices, column matrix, horizontal matrix, and vertical matrix are all rectangular matrices.

### Square Matrix

A matrix where the number of rows is equal to the number of columns is called a square matrix.  A matrix $\text{A} = [a_{ij}]_{m \times n}$ is called a square matrix, if $m = n$.

#### Example of a Square Matrix

Example 1

The matrix $\text{M}$ is a square matrix, with a number of rows equal to the number of columns which is $4$.

### Diagonal Matrix

A square matrix is called a diagonal matrix, if all the elements of the matrix, except the principal diagonal elements, are all equal to zero. Thus a square matrix $\text{A} = [a_{ij}]_{m \times m}$  is a diagonal matrix if $a_{ij} = 0$, when $i \ne j$

#### Examples of Diagonal Matrix

Example 1

In the above matrix $\text{D}$, all the elements are $0$, except the diagonal elements. The diagonal elements are $d_{1, 1} = 2$, $d_{2, 2} = 4$, $d_{3, 3} = -1$, and $d_{4, 4} = 9$.

Note: A diagonal matrix is always a square matrix.

### Scalar Matrix

A diagonal matrix is called a scalar matrix if all the diagonal elements are equal. Thus, a square matrix $\text{A} = [a_{ij}]_{m \times m}$, is a scalar matrix, if $a_{ij} = k$, when $i \ne j$, where $k$ is a non-zero number.

#### Examples of Scalar Matrix

Example 1

In the above matrix $\text{S}$, all the elements are $0$, except the diagonal elements, and are unequal. The diagonal elements are $s_{1, 1} = 2$, $s_{2, 2} = 2$, $s_{3, 3} = 2$, and $s_{4, 4} = 2$.

In the above matrix, the scalar $k = 2$.

Note:

• A scalar matrix is always a diagonal matrix
• A scalar matrix is always a square matrix

### Unit or Identity Matrix

A diagonal matrix is called a unit or identity matrix if all the elements of the principal diagonal $1$. A unit matrix of order $m \times n$ can be denoted by $\text{I}_n$. Thus, a square matrix $\text{A} = [a_{ij}]_{m \times m}$ is an identity matrix if all its diagonal elements have a value equal to $1$.

#### Examples of Unit or Identity Matrix

Example 1

The above matrix is a unit matrix of order $3$ and is represented as $\text{I}_3$.

Example 2

The above matrix is a unit matrix of order $4$ and is represented as $\text{I}_4$.

### Zero or Null Matrix

A matrix where all the elements are zero then it is called a zero matrix or a null matrix and it is generally denoted by $\text{O}$. Thus, a matrix $\text{A} = [a_{ij}]_{m \times n}$ is called a zero matrix, if $a_{ij} = 0$, for all $i$ and $j$.

#### Examples of Zero or Null Matrix

Example 1

The above matrix is a zero or null matrix of order $3$ and is represented as $\text{O}_3$.

Example 2

The above matrix is a zero or null matrix of order $4$ and is represented as $\text{O}_4$.

## Practice Problems

1. What is a matrix?
2. Explain the following with examples
• Row Matrix
• Column Matrix
• Horizontal Matrix
• Vertical Matrix
• Rectangular Matrix
• Square Matrix
• Zero or Null Matrix
• Diagonal Matrix
• Scalar Matrix
• Unit or Identity Matrix

## FAQs

### What are different types of matrices?

The different types of matrices are Row Matrix, Column Matrix, Horizontal Matrix, Vertical Matrix, Rectangular Matrix, Square Matrix, Zero or Null Matrix, Diagonal Matrix, Scalar Matrix, and Unit or Identity Matrix.

### What type of matrix is a 3×3?

A 3×3 is a square matrix. Any matrix of order $m \times m$ is a square matrix.

### What type of matrix is a 3×4?

A 3×4 is a rectangular matrix. Any matrix of order $m \times n$ is a rectangular matrix. It is also called a horizontal matrix as the number of columns is greater than the number of rows.

### What type of matrix is a 4×3?

A 4×3 is a rectangular matrix. Any matrix of order $m \times n$ is a rectangular matrix. It is also called a vertical matrix as the number of rows is greater than the number of columns.

### Can a rectangular matrix be a diagonal matrix?

No, a rectangular matrix cannot be a diagonal matrix. A diagonal matrix is always a square matrix.

## Conclusion

A matrix in math is an ordered rectangular array of numbers consisting of $m$ horizontal rows and $n$ vertical columns. There are various types of matrices. The most common types of matrices are Row Matrix, Column Matrix, Horizontal Matrix, Vertical Matrix, Rectangular Matrix, Square Matrix, Zero or Null Matrix, Diagonal Matrix, Scalar Matrix, and Unit or Identity Matrix.