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Symmetric and Skew Symmetric Matrix

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Last updated date: 01st Mar 2024
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What is Symmetric and Skew Symmetric Matrix?

Matrix: Matrix in mathematics is defined as an array of numbers arranged in a rectangular fashion and divided between rows and columns. It contains all the numbers arranged in square brackets. The operation of matrices is a very important topic in mathematics for the board exams as well as for the engineering entrance exams.


What is a Symmetric Matrix?

  • A square matrix that is equal to its transpose is known as a symmetric matrix.

  • Only square matrices are symmetric because only equal matrices have equal dimensions.

  • A matrix A with nn dimensions is said to be skew-symmetric if and only if 

aij = aji for all i, j such that 1≤n, j≤n.


  • Suppose A is a matrix, then if the transpose of matrix A = AT is equal, it is a symmetric matrix.

  • Symmetric matrix example,

                 A     =   \[\begin{bmatrix}1 & 1 & -1\\ 1 & 2 & 0\\ -1 & 0 & 5\end{bmatrix}\]


            The transpose of A (AT) = \[\begin{bmatrix}1 & 1 & -1\\ 1 & 2 & 0\\ -1 & 0 & 5\end{bmatrix}\]


       Since, A= AT matrix A is a symmetric matrix.


NOTE: If any diagonal matrix is equal to the transpose of the matrix, such matrices are automatically symmetric.


Before We Move Further, Let Us Know About Some Important Terms!


  • A square matrix is a matrix where the number of columns is equal to the number of rows.


If m=n, the matrix is a square matrix

If m ≠ n, the matrix is a rectangular matrix


Here, m = The number of rows


n = The number of columns


Skew Symmetric Matrix Definition

  • A square matrix is said to be skew-symmetric if the transpose of the matrix equals its negative.

  • A matrix A with nn dimensions is said to be skew-symmetric if and only if 

aij = -aji for all i, j such that 1≤n, j≤n.


  • Suppose A is a matrix, then if the transpose of matrix A, AT =- A is equal then it is a skew-symmetric matrix.


First, let us know how to find the Transverse of a Matrix


Transpose of a Matrix (AT)

We find the transpose of a matrix by interchanging the rows and columns of the original matrix. Suppose the original matrix is denoted by n×m, the transpose of the matrix will be m×n.


Let us take an example, 


If A=  \[\begin{bmatrix}1 & 2\\ 3 & 4\end{bmatrix}\], then let us calculate the transpose of the matrix A.


Here, the first row becomes the first column and the second row becomes the second column.


AT= \[\begin{bmatrix}1 & 3\\ 2 & 4\end{bmatrix}\]


Here we see that A AT.


If A =  \[\begin{bmatrix}1 & 1 & -1\\ 1 & 2 & 0\\ -1 & 0 & 5\end{bmatrix}\], then let us calculate the transpose of the matrix A.


Here, the first row becomes the first column, the second row becomes the second column and the third row becomes the third column.


AT = \[\begin{bmatrix}1 & 1 & -1\\ 1 & 2 & 0\\ -1 & 0 & 5\end{bmatrix}\]


Here, we see that A = AT


How to check Whether a Matrix is Symmetric or Not?

Step 1- Find the transpose of the matrix.


Step 2- Check if the transpose of the matrix is equal to the original matrix.


Step 3- If the transpose matrix and the original matrix are equal, then the matrix is symmetric.


Example 1


              A =  \[\begin{bmatrix}0 & 2 & -45\\ -2 & 0 & -4\\ 45 & 4 & 0\end{bmatrix}\]


            - A=  \[\begin{bmatrix}0 & -2 & 45\\ 2 & 0 & 4\\ -45 & -4 & 0\end{bmatrix}\]


                  = AT


                Since, AT=-A matrix A is a skew-symmetric matrix


Example 2

             P =  \[\begin{bmatrix}0 & -5\\ 5 & 0\end{bmatrix}\]


            -P =  \[\begin{bmatrix}0 & 5\\ -5 & 0\end{bmatrix}\]


                 = PT


              Since, PT=-P matrix A is a skew-symmetric matrix.


Conditions for Symmetric and Skew Symmetric Matrix 


Symmetric Matrix(A)

AT=A

aji=(aij)

Skew Symmetric Matrix (A)

AT=(-A)

aji=(-aij)


Here, i = Row entry


j = Column entry


How to check whether a Matrix is Skew Symmetric or not?

Step 1 - First find the transpose of the originally given matrix.


Step 2 – Then find the negative of the original matrix.


Step 3 – If the negative of the matrix obtained in Step2 is equal to the transpose of the matrix then the matrix is said to be skew-symmetric.


Properties:

Any matrix A can be written as a sum of /symmetric matrix and a skew-symmetric matrix.


\[A=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')\]


Questions to solve

Question 1:  Check whether the given matrices are symmetric or not.


  • M =   \[\begin{bmatrix}0 & 5\\ 9 & 0\end{bmatrix}\]


  • P =   \[\begin{bmatrix}1 & 4\\ 0 & -1\end{bmatrix}\]  


Solution: We will first find the transpose of matrix M,


MT = \[\begin{bmatrix}0 & 9\\ 5 & 0\end{bmatrix}\]


Since the transpose of M is not equal to matrix M, therefore it is not a symmetric matrix. 

We will first find the transpose of matrix P,


PT = \[\begin{bmatrix}1 & 0\\ 4 & -1\end{bmatrix}\]


Since the transpose of P is not equal to matrix P, therefore it is not a symmetric matrix. 


Question 2 : Is the given matrix A, a skew-symmetric matrix. Give a reason for your answer.


\[A=\begin{bmatrix}0 & -1\\ 1 & -0\end{bmatrix}\]


Solution: First, we will find the transpose of the matrix A,


\[A^T=\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}\]


Now we will find the negative of the matrix A.


\[-A=\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}\]


Since, the negative of the matrix A is equal to the transpose of the matrix A. 


Therefore, A is a skew-symmetric matrix.


Question 3: Show that the given matrix is a symmetric matrix.


                   \[A=\begin{bmatrix}1 & 2 & 3\\ 2 & 4 & 5\\ 3 & 5 & 8\end{bmatrix}\]  


Solution: To check whether the given matrix A is a symmetric matrix,


We need to find the transpose of the given matrix A,


\[A^T=\begin{bmatrix}1 & 2 & 3\\ 2 & 4 & 5\\ 3 & 5 & 8\end{bmatrix}\]


Since the original matrix A is equal to the transpose matrix, therefore the given matrix A is a symmetric matrix.


Question 4: Check whether the given matrix B is a symmetric matrix or a skew-symmetric matrix.

          

\[B=\begin{bmatrix}0 & 5 & 3\\ -5 & 0 & -8\\ -3 & 8 & 0\end{bmatrix}\]


Solution: Let’s check whether the given matrix is symmetric or not.

We need to find the transpose of the given matrix B,


\[B^T=\begin{bmatrix}0 & -5 & -3\\ 5 & 0 & 8\\ 3 & -8 & 0\end{bmatrix}\]


Since the original matrix B is not equal to the transpose matrix (BT≠B), therefore the given matrix B is not a symmetric matrix.


Let’s check whether the given matrix is skew-symmetric or not.


Since we have already found the transpose,


\[B^T=\begin{bmatrix}0 & -5 & -3\\ 5 & 0 & 8\\ 3 & -8 & 0\end{bmatrix}\]


We will find the negative of the original matrix.


\[-B=\begin{bmatrix}0 & -5 & -3\\ 5 & 0 & 8\\ 3 & -8 & 0\end{bmatrix}\]


Since the negative of the matrix B is equal to the transpose of the matrix B. 

Therefore, B is a skew-symmetric matrix.


Fun Facts about Matrices:

  • The term “Matrix” was coined by James Sylvester who was a 19th-century British Mathematician

  • The algebra of Matrices was developed by the mathematician Arthur Cayley who was James Sylvester’s friend.


Applications of Matrices:

Very often students studying Maths keep wondering about the practical applications of the concept. Thus, to do away with this wonderment, Vedantu has brought a list of practical and real-life applications of Matrices. These are as follows:  


  • Matrices are used to represent real-world data like the population of people, maternal mortality rate, etc

  • Medical imaging, MRIs use matrices to operate

  • Matrices also have applications in creating video games

  • Electronics networks, aeroplanes and spacecraft, all require well-calibrated computations that are obtained from matrix transformations

  • Matrices are also used in the field of Economics

  • Matrix in computer graphics is used to convert geometric data into different coordinate systems.

  • Matrices are used in business as well. A decision matrix is used to make help the user make a complex decision 

FAQs on Symmetric and Skew Symmetric Matrix

1. What is a Symmetric matrix? Give an example.

A square matrix B which of size n × n is considered to be symmetric if and only if BT = B. Consider the given matrix B, that is, a square matrix that is equal to the transposed form of that matrix, called a symmetric matrix.

Example is given below,


\[B=\begin{bmatrix}2 & 3 & 6\\ 3 & 4 & 5\\ 6 & 5 & 9\end{bmatrix}\]


\[B^T=\begin{bmatrix}2 & 3 & 6\\ 3 & 4 & 5\\ 6 & 5 & 9\end{bmatrix}\]

2. Is the zero-matrix symmetric?

Yes, Zero - matrix is symmetric

3.What are symmetric and skew symmetric matrices?

Symmetric Matrix

For a matrix B, if B = B’ (Matrix B = Transpose of Matrix B), that is whenever the transpose of a matrix is equal to it, the matrix is known as a symmetric matrix.


\[\begin{bmatrix}1 & 2 & 3\\ 2 & 4 & 5\\ 3 & 5 & 8\end{bmatrix}=\begin{bmatrix}1 & 2 & 3\\ 2 & 4 & 5\\ 3 & 5 & 8\end{bmatrix}^T\]


Skew-Symmetric Matrix

A skew-symmetric matrix (also known as antisymmetric or antimetric) is a square matrix whose transpose equals the negative of the matrix.


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4. Can we diagonalize a Symmetric Matrix?

Yes, we can diagonalize a symmetric Matrix

5. How many questions are asked from the topic Symmetric and Skew Symmetric Matrix in JEE Main exams?

You can expect 1-2 questions in the JEE Main exam from Symmetric and Skew Symmetric Matrix. Each question is of 4 marks. Thus you can expect questions worth 8 marks in the JEE Mains. These many marks can play a very decisive role in deciding whether you will take one step ahead towards your dream IIT or many steps back from it. Therefore it is advised you should not take this topic for granted and practise enough questions to master this topic.    

6. What is the weightage of the topic Symmetric and Skew Symmetric Matrix in Class 12 Boards exams?

The Class 12th Board exam does not have chapter wise marks distribution as such. But going by the past trends you can expect 5-8 marks worth of questions from matrices. One of these questions may be from Symmetric and Skew Symmetric Matrix. You cannot risk not knowing this topic in and out. You need to practice enough questions after completing the topic from Vedantu’s website. You can find solutions to NCERT Class 12 Maths Chapter 3 Matrices on the website. We suggest you solve all the problems given in the chapter.   

7. How many sums do I need to solve to master the topic of Symmetric and Skew Symmetric Matrices?

There is no fixed number of sums that can guarantee that solving them will make you a master in the topic of Symmetric and Skew Symmetric Matrix. You should learn this topic from the very basics of matrices. After this solve questions given in Chapter 3 of NCERT Class 12 Maths textbook. After this, you should proceed towards solving matrices questions from RD Sharma’s Class 12 book. Solutions for RD Sharma Class 12 Maths Solutions Chapter 5 - Algebra of Matrices can be found on Vedantu’s website for free. Solving these many problems will help you form a command over this topic and you will be in a position to solve any question from this topic.

8. Can I understand the topic of Symmetric and Skew Symmetric Matrices merely by going through Vedantu’s website?

Yes, you can understand the topic just by referring to Vedantu’s website. After going through the website, you will learn the meaning of symmetric and skew-symmetric matrices, the difference between them, identifying whether a given matrix is symmetric or skew-symmetric, conditions of symmetry and skew-symmetry. You will also learn to find the transpose of a matrix. Towards the end, four questions have been for you to practice and test your understanding. Vedantu suggests you should practice additional problems from the Class 12 Maths NCERT and RD Sharma’s Class 12 Maths books.    

9. What concepts do I need to know before starting the topic of Symmetric and Skew Symmetric Matrices?

Before starting with the topic of Symmetric and Skew Symmetric Matrix you need to know some topics in advance. You need to be thorough with the definition of a Matrix and various types of matrices such as 2x2 matrix, 3x3 matrix, etc. Apart from this, you should also know how to find the Transpose of a matrix. You can refer to Vedantu's website to know more about the transpose of a matrix. You are also expected to know certain properties of matrices beforehand. Once you know these many concepts you will be prepared to tackle any question based on Symmetric and Skew Symmetric Matrix.