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.

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 

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