For any square matrix A,

  (A + A’) is a symmetric matrix

  (A − A’) is a skew-symmetric matrix

 

Let’s first prove them

 

(A + A’) is a symmetric matrix

 

For a symmetric matrix

  X’ = X

So, we have to prove

   (A + A’)’ = (A + A’)

 

Solving LHS

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Therefore,

  (A + A’)’ = A + A’

So, A + A’ is a symmetric matrix

 

(A − A’) is a symmetric matrix

 

For a skew symmetric matrix X

  X’ = −X

So, we have prove

      (A − A’)’ = − (A − A’)

 

Solving LHS

83.jpg

Therefore,

  (A − A’)’ = − (A − A’)

 

So, A − A’ is a skew symmetric matrix

 

Now,

Let’s write matrix A as sum of symmetric & skew symmetric matrix

       (A + A’) + (A − A’) = 2A

So,

  1/2 [(A + A’) + (A − A’)] = A

  1/2 (A + A’) + 1/2 (A − A’) = A

84.jpg

Here,

      1/2 (A + A’) is the symmetric matrix

&   1/2 (A − A’) is the symmetric matrix

 

Let’s take an example,

85.jpg

 

Let’s check if they are symmetric & skew-symmetric

86.jpg

 

  1. Chapter 3 Class 12 Matrices
  2. Concept wise

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.