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[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] - [A]T, respectively. Which of the following statements is true?
1. Both [S] and [D] are symmetric
2. Both[S] and [D] are skew-symmetric
3. [S] is skew-symmetric and [D] is symmetric
4. [S] is symmetric and [D] is skew-symmetric
5.

1 Answer

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Correct Answer - Option 4 : [S] is symmetric and [D] is skew-symmetric

Explanation:

Given,

[A] is a square matrix and it is neither symmetric nor skew-symmetric

Also given,

[AT]T [A]

[S] = [A] + [A]T , [D] = [A] - [A]T

Lets take  Case (i)

[S]T = ( [A] + [A]T )T = [A]T + A = [S]

So [S] is a Symmetric Matrix

Now Case (ii)

[D]T = ([A] - [A]T)T = [A]T - [AT]T = [A]- [A] = - ([A] - [A]T) = - [D]

So [D] is skew Symmetric Matrix

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