# Types of Matrices

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## What is a Matrix?

Before discussing the types of matrix, let’s discuss what a matrix is.

• A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.

• The order of the matrix is defined as the number of rows and columns.

• The entries are the numbers in the matrix and each number is known as an element.

• The plural of matrix is matrices.

• The size of a matrix is referred to as ‘n by m’ matrix and is written as m × n, where n is the number of rows and m is the number of columns.

• Matrix example, we have a 3 × 2 matrix, that’s because the number of rows here is equal to 3 and the number of columns is equal to 2.

$\begin{bmatrix}-2&5&6 \\ 5&2&7 \end{bmatrix}$

The dimensions of a matrix can be defined as the number of rows and columns of the matrix in that order. Since the matrix A given above has 2 rows and 3 columns, it is known as a 2 × 3 matrix.

### What are the Different Types of Matrices?

There are different types of Matrices. Here they are –

1) Row matrix

2) Column matrix

3) Null matrix

4) Square matrix

5) Diagonal matrix

6) Upper triangular matrix

7) Lower triangular matrix

8) Symmetric matrix

9) Skew -symmetric matrix

10) Horizontal matrix

11) Vertical matrix

12) Identity matrix

Let’s discuss the different types of matrices in mathematics, types of matrix in detail, matrices definition and types.

### 1. What is a Null Matrix?

If in a matrix all the elements are zero then it is called a zero matrix and it is generally denoted by 0. Thus, A = [$a_{ij}$] m x n is a zero-matrix if $a_{ij}$= 0 for all i and j.

The first matrix O is a 2 × 2 matrix with all the elements equal to zero and the second matrix O is a 3 × 3 matrix with all the elements equal to zero.

O = $\begin{bmatrix}0&0 \\ 0&0 \end{bmatrix}$ , O = $\begin{bmatrix}0&0&0 \\ 0&0&0 \\ 0&0&0 \end{bmatrix}$

### 2. What is a Triangular Matrix?

A square matrix is known to be triangular if all of its elements above the principal diagonal are zero then the triangular matrix is known as a lower triangular matrix or all of its elements below the principal diagonal are zero then the triangular matrix is known as upper triangular matrix).

$\begin{bmatrix}1&0&0 \\ 2&4&0 \\ 3&5&6 \end{bmatrix}$

The matrix given above is a 3 × 3 upper triangular matrix.

The matrix given below is an example of a 3 × 3 lower triangular matrix.

$\begin{bmatrix}1&2&3 \\ 0&6&5 \\ 0&0&9 \end{bmatrix}$

### 3. What is  a Vertical Matrix?

A matrix of order m x n is known as a vertical matrix if m > n, where m is equal to the number of rows and n is equal to the number of columns.

Matrix Example

$\begin{bmatrix}2&5 \\ 1&1 \\ 3&6 \\ 2&4 \end{bmatrix}$

In matrix example given below the number of rows (m) = 4 , whereas the number of columns (n) = 2. Therefore, this makes the matrix a vertical matrix.

### 4. What is  a Horizontal Matrix?

A matrix of order m x n is known as a horizontal matrix if n > m, where m is equal to the number of rows and n is equal to the number of columns.

Matrix Example

$\begin{bmatrix}1&2&3&4 \\ 2&5&1&1 \end{bmatrix}$

In the matrix example given below the number of rows (m) = 2 , whereas the number of columns (n) = 4. Therefore, we can say that the matrix is a horizontal matrix.

### 5. What is a Row Matrix?

A matrix that has only one row is known as a row matrix. Thus A = [$a_{ij}$] m x n is a row matrix if m is equal to 1. So, a row matrix can be represented as A = [$a_{ij}$] 1 × n. It is known so because it has only one row and the order of a row matrix will hence always be equal to 1 × n.

Example of  a Row matrix,

A = $\begin{bmatrix}4&6&9 \end{bmatrix}$ , B = $\begin{bmatrix}7&2&1&9&2&5 \end{bmatrix}$

In matrix example given above, matrix A has only one row and so matrix B has one row, therefore both matrices A and B are row matrices.

### 6. What is a Column Matrix?

A matrix that has one column is known as a Column matrix. Thus A = [$a_{ij}$]m x n is a column matrix if n is equal to 1. So, a row matrix can be represented as A = [$a_{ij}$]m × 1. It is known so because it has only one column and the order of a column matrix will hence always be equal to m × 1.

Example of  a Column matrix,

A =  $\begin{bmatrix}3 \\ 4 \\ 8 \end{bmatrix}$, B =  $\begin{bmatrix}4 \\ 9 \\ 8 \\ 2 \end{bmatrix}$

In matrix example given above, matrix A has only one column and matrix B has one column, therefore both matrices A and B are column matrices.

### 7. What is a Diagonal Matrix?

If all the elements of the matrix, except the principal diagonal in any given square matrix, is equal to zero, it is known as a diagonal matrix. Thus,a square matrix A = [$a_{ij}$] is a diagonal matrix if $a_{ij}$ = 0, when i is not equal to j.

For example,

$\begin{bmatrix}2&0&0 \\ 0&3&0 \\ 0&0&4 \end{bmatrix}$

The example given above is a diagonal matrix as it has elements only in its diagonal.

### 8. What is a Symmetric Matrix?

A square matrix A = [$a_{ij}$] is known as a Symmetric matrix if $a_{ij}$ = $a_{ji}$, for all i,j values.

For example,

A = $\begin{pmatrix}1&2&3 \\ 2&4&5 \\ 3&5&2 \end{pmatrix}$

### 9. What is the Skew -Symmetric Matrix?

A square matrix A = [$a_{ij}$] is a skew-symmetric matrix if $a_{ij}$ = $a_{ji}$, for all values of i,j. Thus, in a skew-symmetric matrix all diagonal elements are equal to zero.

For example,

$\begin{bmatrix}0&2&1 \\ -2&0&-3 \\ -1&3&0 \end{bmatrix}$

### 10. What is an Identity Matrix?

If all the elements of a principal diagonal in a diagonal matrix are 1, then it is called a unit matrix. A unit matrix of order n can be denoted by In. Thus, a square matrix A = [ $a_{ij}$] m × n is an identity matrix if all its diagonals have value 1.

For example,A = $\begin{bmatrix}1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix}$

### Questions to be Solved

Question 1) Give an example of an identity matrix with a number of rows and columns equal to two.

Answer) We know that an identity matrix is one with its diagonal elements equal to 1 and all other elements equal to zero.

For example,A = $\begin{bmatrix}1&0 \\ 0&1 \end{bmatrix}$

Question 1) What is a Matrix?What are the types of Matrices with examples?

Answer) Matrix refers to a rectangular array of numbers. A matrix consists of rows and columns. The Types Of Matrices are-

• A matrix that has only one row is known as a row matrix.

• A matrix that has only one column is known as a column matrix.

• A vector matrix is a column matrix that is of order 2 ×1 .

• A zero matrix or a null matrix is a matrix that has all its elements equal to zero.

Question 2) Who is the father of Matrices?What are the applications of Matrix?

Answer) An English mathematician and lawyer named Arthur Cayley (1821-1895), who was the first one to publish an abstract definition of a matrix in his Memoir on the Theory of Matrices in the year 1858, thus establishing it as a branch of mathematics. So this man was the father of the matrix. Applications of matrices can be easily found in most scientific fields. Matrices have their use in every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics; they are used to study physical phenomena, such as the motion of rigid bodies.