HOW TO EXPRESS THE GIVEN MATRIX AS SUM OF SYMMETRIC AND SKEW SYMMETRIC

About "How to Express the Given Matrix as Sum of Symmetric and Skew Symmetric"

How to Express the Given Matrix as Sum of Symmetric and Skew Symmetric :

Here we are going to see how to express the given matrix as the sum of symmetric and skew symmetric matrix.

What is symmetric and skew symmetric matrix ?

For any square matrix A with real number entries, A+ A^T is a symmetric matrix and A− A^T is a skew-symmetric matrix.

Any square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Let A be a square matrix. Then, we can write

Let us look into some problems to understand the concept.

Question 1 :

Express the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix:

Solution :

First let us add the matrices A and A^T, then we have to multiply it by 1/2

Now we have to subtract the matrices A and A^T, then we have to multiply it by 1/2

By adding the above two matrices, we get the original question.

Hence proved.

(ii)  From the given matrix A, we have to find A^T

So far we get symmetric matrix. Now we are going to find skew symmetric matrix.

By adding (1) and (2), we get

Hence proved.

Question 3 :

Find the matrix A such that

Solution :

Let the required matrix A^T be

By equating the corresponding terms, we get

2a-d  =  -1  ----(1)

2b - e  = -8 ----(2)

2c - f  =  -10 ----(3)

a  =  1, b  =  2  and c  =  -5

By applying the value of a in the first equation, we get the value d.

2(1) - d  =  -1

 2 - d  =  -1

-d  =  -1 - 2

-d  =  -3

d  =  3

By applying the value of b in the second equation, we get the value e.

2(2) - e  = -8

 4 - e  =  -8

-e  =  -8 - 4

-e  =  -12

e  =  12

By applying the value of c in the third equation, we get the value f.

2(-5) - f  =  -10

 -10 - f  =  -10

-f  =  -10 + 10

f  =  0

After having gone through the stuff given above, we hope that the students would have understood "How to Express the Given Matrix as Sum of Symmetric and Skew Symmetric".

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