Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where A = [52-43-724-5-3] - Mathematics and Statistics

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Sum

Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where A = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)]`

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Solution

A = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)]`

∴ AT = `[(5, 3, 4),(2, -7, -5),(-4, 2, -3)]`

∴ A + AT = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)] + [(5, 3, 4),(2, -7, -5),(-4, 2, -3)]`

= `[(5 + 5, 2 + 3, -4 + 4),(3 + 2, -7 - 7, 2 - 5),(4 - 4, -5 + 2, -3 - 3)]`

∴ A + AT = `[(10, 5, 0),(5, -14, -3),(0, -3, -6)]`

∴  (A + AT)T = A + AT i.e., A + AT = (A + AT)T 
∴ A + AT = is a symmetric matrix.

A – AT = `[(5, 2, -4),(3, -7, 2),(4, -5, -3)] - [(5, 3, 4),(2, -7, -5),(-4, 2, -3)]`

= `[(5 - 5, 2 - 3, -4 - 4),(3 - 2, -7 + 7, 2 + 5),(4 + 4, -5 - 2, -3 + 3)]`

= `[(0, -1, -8),(1, 0, 7),(8, -7,0)]`

∴ (A – AT)T = `[(0, 1, 8),(-1, 0, -7),(-8, 7,0)]`

= `[(0, -1, -8),(1, 0, 7),(8, -7,0)]`

∴ (A – AT)T = – (A – AT)
i.e., A – AT = – (A – AT)T
∴ A – AT  is a skew symmetric matrix.

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Chapter 2: Matrices - Exercise 2.4 [Page 59]

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