Express the following matrix as the sum of a symmetric and a skew symmetric matrix [4-23-5] - Mathematics and Statistics

Question

Express the following matrix as the sum of a symmetric and a skew symmetric matrix

`[(4, -2),(3, -5)]`

Sum

Solution

For any matrix A, A + A' is always symmetric and A – A' is always skew-symmetric.

∴ A = `1/2[("A" + "A""'") + ("A" - "A""'")]`

i.e., A = `1/2("A" + "A""'") + 1/2("A" - "A""'")`

Here, A = `[(4, -2),(3, -5)]`

∴ A' = `[(4, 3),(-2, -5)]`

∴ A + A' = `[(4, -2),(3, -5)] + [(4, 3),(-2, -5)]`

= `[(8, 1),(1, -10)]`

and A – A' = `[(4, -2),(3, -5)] - [(4, 3),(-2, -5)]`

= `[(0, -5),(5, 0)]`

∴ A = `1/2[(8, 1),(1, -10)] + 1/2[(0, -5),(5, 0)]`

i.e., A = `[(4, 1/2),(1/2, -5)] + [(0, (-5)/2),(5/2, 0)]`

i.e., A is the sum of the symmetric matrix `[(4, 1/2),(1/2, -5)]` and the skew-symmetric matrix `[(0, (-5)/2),(5/2, 0)]`.

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Q 12. (i)Q 11. (ii)Q 12. (ii)

Chapter 4: Determinants and Matrices - Exercise 4.7 [Page 98]

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Balbharati Mathematics and Statistics 1 (Arts and Science) [English] 11 Standard Maharashtra State Board

Chapter 4 Determinants and Matrices
Exercise 4.7 | Q 12. (i) | Page 98

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