# Types of Matrices

Updated: 07 Mar 2023

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For a beginner, understanding the different types of matrices and their uses can take time and effort. That’s why we’ve put together this beginner’s guide to matrices, which will help you understand the basics of matrices and their various types. We’ll cover everything from square matrices to identity matrices, zero matrices, diagonal matrices, and more. We’ll also provide examples of each type of matrix to understand you in a better way. By the end of this guide, you’ll have a solid foundation in matrices and be able to tackle more complex problems easily. So, let’s get started and explore the types of matrices together!

## What is a Matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is denoted by a capital letter, and the elements of a matrix are represented by lowercase letters. The number of rows and columns determines the size of a matrix. For example, a matrix with m rows and n columns is said to be an m x n matrix. Matrices are used to represent data in many different ways. For example, a matrix can represent a system of linear equations, where each row represents an equation, and each column represents a variable.

## Different Types of Matrices

Learn about the various types of matrices, including Square, Diagonal, Identity, Zero, Transpose, Symmetric, and Skew-symmetric matrices.

### Row Matrix

##### What is a Row Matrix
A matrix having just one row is called row matrix.
Examples of Row Matrix
A=\left[\begin{array}{lll}1 & 3 & 5\end{array}\right]
B=[5]

جس میٹرکس میں صرف ایک Row ہو اسے Row Matrix کہا جاتا ہے۔

### Column Matrix

##### What is a Column Matrix
A matrix having just one column is called column matrix.
Examples of Column Matrix
A=\left[\begin{array}{l}1 \\ 3 \\ 5\end{array}\right]
B=[5]

جس میٹرکس میں صرف ایک Column ہو اسےColumn Matrix کہا جاتا ہے۔

### Square Matrix

##### What is Square Matrix
A matrix in which number of rows and columns are equal is called square matrix.
Examples of Square Matrix
A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 5 \end{array}\right]
B=[5]
C=\left[\begin{array}{ccc} 1 & 3 & 2 \\ -2 & 0 & 4 \\ -1 & 3 & 6 \end{array}\right]

ایک میٹرکس جس میں Rows اور Columns کی تعداد برابر ہو اسے Square Matrix کہا جاتا ہے۔

### Rectangular Matrix

##### What is Rectangular Matrix
A matrix in which number of rows and columns are not equal is called rectangular matrix.
Examples of Rectangular Matrix
A=\left[\begin{array}{l} 1 \\ 3 \\ 5 \end{array}\right]
D=\left[\begin{array}{lll} 3 & 5 & 2 \\ 0 & 9 & 8 \end{array}\right]

ایک میٹرکس جس میں Rows اور Columns کی تعداد برابرنہ ہو اسے Rectangular Matrix کہا جاتا ہے۔

### Zero Matrix or Null Matrix

##### What is Zero Matrix or Null Matrix
A matrix in which all the elements are zero is called Zero or Null matrix. A null matrix is generally denoted by O.
Examples of Zero Matrix
O=\left[\begin{array}{l} 0 \\ 0 \end{array}\right]
O=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]
O=\left[\begin{array}{lll} 0 & 0 & 0 \end{array}\right]

ایک میٹرکس جس میں تمام Elements ہمارے ساتھ Zero ہو اسے Zero یا Null کہتے ہیں۔ ایک Null Matrix کو عام طور پر O سے ظاہر کیا جاتا ہے۔

### Diagonal Matrix

##### What is Diagonal Matrix
A square matrix in which all elements are zero except diagonal elements is known as diagonal matrix.
Examples of Diagonal Matrix
A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 3 \end{array}\right]
B=\left[\begin{array}{cc} -2 & 0 \\ 0 & -2 \end{array}\right]
I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]

Square Matrix جس میں Diagonal کے علاوہ تمام Elements ہمارے ساتھ Zero ہو اسے Diagonal Matrix کہتے ہیں۔

### Scalar Matrix

##### What is Scalar Matrix?

A matrix in which diagonal elements are same is called scalar matrix.
Examples of Scalar Matrix
I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]
A=\left[\begin{array}{lll}7 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 7\end{array}\right]
C=\left[\begin{array}{cc}\frac{1}{2} & 0 \\ 0 & \frac{1}{2}\end{array}\right]
Note
Every scalar matrix is a diagonal matrix but every diagonal matrix is not necessarily a scalar matrix.

Square Matrix جس میں Diagonal کے Elements ہمارے ساتھ ایک جیسے ہو اسے Scalar Matrix کہتے ہیں۔

### Identity Matrix or Unit Matrix

##### What is Identity Matrix or Unit Matrix
A matrix in which the diagonal elements are equal to “1” is called identity matrix. It is generally denoted by “I”.
Examples of Identity Matrix
I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]
I=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]

Square Matrix جس میں Diagonal کے Elements ہمارے ساتھ صرج “1” ہو اسے Scalar Matrix کہتے ہیں۔

### Transpose of a Matrix

##### What is Transpose of a Matrix
A matrix obtained by interchanging all rows and columns with each other is called transpose of a matrix. The transpose of a matrix B is written as B^t.
Example of Transpose Matrix
A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]
A^{t}=\left[\begin{array}{ll}a & c \\ b & d\end{array}\right]

تمام Rows اور Columns کو ایک دوسرے کے ساتھ تبدیل کرکے حاصل کردہ میٹرکس کو Transpose of a Matrix کہا جاتا ہے۔ میٹرکس B کا ٹرانسپوز B^t لکھا جاتا ہے۔

### Symmetric matrix

##### What is Symmetric matrix
In a square matrix, when A^t=A, then A is said to be symmetric matrix.
Example of Symmetric Matrix
A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6\end{array}\right]
A^{t}=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6\end{array}\right]
A^{t}=A

جب کسی Matrix کا Transposeلینے کے بعد دوبارہ وہی Matrixآجا ئے تو اسے Symmetric Matrix کہتے ہیں۔

### Skew-Symmetric Matrix

##### What is Skew-Symmetric Matrix
In a square matrix, when A^t=-A, then A is said to be skew-symmetric matrix.
Examples of Skew-Symmetric Matrix
A=\left[\begin{array}{cc}0 & 4 \\ -4 & 0\end{array}\right]

A^{t}=\left[\begin{array}{cc}0 & -4 \\ 4 & 0\end{array}\right]
A^{t}=-\left[\begin{array}{cc}0 & 4 \\ -4 & 0\end{array}\right]
A^t=-A

جب کسی Matrix کا Transposeلینے کے بعد اور لینے Minus کے بعد دوبارہ وہی Matrixآجا ئے تو اسے Symmetric Matrix کہتے ہیں۔

## Types of Matrices MCQs with Explanation

Welcome to the world of matrices! If you learn the basics of MCQs related to Types of Matrices, this page will help you and provide the best material. We’ll give detailed explanations to help you understand the concepts and improve your problem-solving skills related to types of matrices.

1. \left[\begin{array}{lll}1 & 3 & 5\end{array}\right] is ______ matrix.
O Row
O Column
O Square
O None

Row
Explanation:
Because the matrix consists of only one row.

2. \left[\begin{array}{lll}1 \\ 3 \\ 5\end{array}\right] is ______ matrix.
O Row
O Column
O Square
O None

Column
Explanation:
Because the matrix consists of only one column.

3. A matrix in which number of Rows equal to number of Columns is called ______ matrix.
O Row
O Column
O Square
O None

Square
Explanation:
Definition of square matrix.

4. \left[\begin{array}{lll}1 & 3 & 5 \\ a & b & c \\ 3 & 5 & 7 \end{array}\right] is ______ matrix.
O Row
O Column
O Square
O None

Square
Explanation:
As number of rows and columns are same. Thus, it is square matrix.

5. \left[\frac{1}{2}\right] is a ______ matrix.
O Row
O Column
O Square
O All of these

All of them
Explanation:
Here the matrix consists of one row and also one column and also same number of row and column which is square matrix. So, all of these is correct option.
\frac{1}{2}

6. [3] is ______ matrix.
O Row
O Column
O Square
O All of them

All of them
Explanation:
Here the matrix consists of one row and also one column and also same number of row and column which is square matrix. So, all of them is correct option.

7. A matrix in which number of row and number of columns are not equal is called _______ matrix.
O Square
O Rectangular
O Null
O None of these

Rectangular
Explanation:
Definition of Rectangular matrix.

8. \left[\begin{array}{lll}1 & 3 & 5 \\ a & b & c \end{array}\right] is ______ matrix.
O Row
O Column
O Square
O Rectangular

Rectangular
Explanation:
As number of rows and columns are not equal, so it is Rectangular matrix.

9. \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right] is ______ matrix.
O Row
O Zero
O Null
O Both b and c

Both b and c
Explanation:
As Zero or null matrix is same. So Both b and c is the correct option.

10. \left[\begin{array}{lll}0 \\ 0 \\ 0\end{array}\right] is ______ matrix.
O Row
O Zero
O Square
O None

Zero
Explanation:
As all the elements are zero in the matrix. That is why, its zero matrix.

11. The zero matrix for \left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right] is ______.
O \left[\begin{array}{l}0 \end{array}\right]
O \left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
O 0
O None of these

\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]
Explanation:
The order zero matrix must be same according to the given matrix which is 2-by-2 .

12. The product of any matrix and zero matrix is a ______
O Identity
O Scalar
O Row
O Zero

Zero
Explanation:
When Zero (0) is multiplied to any number or matrix, the answer must be zero.

13. \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 9 \end{array}\right] is ______ matrix.
O Diagonal
O scalar
O Null
O None of these

Diagonal
Explanation:
A square matrix on which all elements are zero except diagonal elements is known as diagonal matrix.

14. Scalar matrix is a special case of ______ matrix.
O Identity
O Diagonal
O Both a & b
O None of these

Diagonal
Explanation:
If the diagonal elements in diagonal matrix are same, then it is called scalar matrix.

15. \left[\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right] is ______ matrix.
O Diagonal
O scalar
O Null
O None of these

Diagonal
Explanation:
Here all the elements are zero except the diagonal elements.

16. The matrix \left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right] is ______ matrix.
O Row
O Scalar
O Null
O None of these

Scalar
Explanation:
As the diagonal elements are same, so it is called scalar matrix.

17. Identity matrix is represented by ______.
O A
O I
O B
O None

I

18. Identity matrix is also called ______ matrix.
O Transpose
O Unit
O Symmetric
O None of these

Unit

19. \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] is ______ matrix.
O Rectangle
O Row
O Identity
O None

Identity
Explanation:
When the diagonal elements are “1” then matrix is called Identity matrix.

20. I_3 means ______
O \left[\begin{array}{l}1 \end{array}\right]
O \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]
O \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]
O All of them

\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]
Explanation:
I_3 means, Identity matrix having 3 rows and 3 columns.

21. I_3 is the identity matrix of order ______
O 3
O 3 -by-3
O 3 \times 3
O All of them

All of them
Explanation:
I_3 means, Identity matrix of 3 rows and columns. Here all the options show the same 3 rows and columns.

22. A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] is ______ matrix.
O Diagonal
O Scalar
O Identity
O All of these

All of these
Explanation:
It is the diagonal matrix having same diagonal elements whcih is scalar matrix and also the diagonal element is 1 which is Identity matrix.
So all are correct options.

23. \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] is ______ matrix
O Diagonal
O Scalar
O Identity
O All of these

All of these
Explanation:
It is the diagonal matrix having same diagonal elements whcih is scalar matrix and also the diagonal element is 1 which is Identity matrix.
So all are correct options.

24. The Transpose of \left[\begin{array}{ll}1 & -2 \\ 3 & \ 4\end{array}\right] is ______.
O \left[\begin{array}{ll}1 & -2 \\ 3 & \ 4\end{array}\right]
O \left[\begin{array}{ll}1 & 3 \\ -2 & \ 4\end{array}\right]
O \left[\begin{array}{ll}1 & 2 \\ 3 & \ 4\end{array}\right]
O All of these

O \left[\begin{array}{ll}1 & 3 \\ -2 & \ 4\end{array}\right]
Explanation:
When rows and columns are interchanged with each other, then it is called transpose of a matrix.

25. The transpose of \left[\begin{array}{l}1 \\ 2\end{array}\right] is ______.
O \left[\begin{array}{lll}1 & 2 & 3\end{array}\right]
O \left[\begin{array}{ll}1 & 2\end{array}\right]
O \left[\begin{array}{l}2 \\ 1\end{array}\right]
O None

\left[\begin{array}{ll}1 & 2\end{array}\right]
Explanation:
When rows and columns are interchanged with each other, then it is called transpose of a matrix.

26. If A^t=A , then it is a ______ matrix
O Symmetric
O Skew-symmetric
O Diagonal
O None

Symmetric
Explanation:
When the transpose of matrix comes again that matrix is called symmetric matrix.

27. If D^t=-D then it is a ______ matrix.
O Symmetric
O Skew-symmetric
O Diagonal
O None

Skew-symmetric
Explanation:
When the transpose of a matrix is equal to the negative of that matrix.

28. The matrix \left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right] is ______ matrix.
O Symmetric
O Skew-symmetric
O scalar
O Diagonal

Symmetric
Explanation:
Take the transpose of
\left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right]^t
\left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right]
Thus, A^t=A

29. The matrix \left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right] is ______ matrix.
O Symmetric
O Skew-symmetric
O scalar
O Diagonal

Skew-symmetric
Explanation:
\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]
Take the transpose, we get
\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]
-\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]
Thus the matrix is skew-symmetric.