Question

Prove that A + A^T is a symmetric and A – A^T is a skew symmetric matrix, where

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

Answer 1

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

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

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

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

∴ (A + A^T)^T = A + A^T , i.e., A + A^T = (A + A^T)^T 

∴ A + A^T is symmetric matrix.

Also, A – A^T = `[(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 – A^T)^T = `[(0, 1, 8),(-1, 0, -7),(-8, 7, 0)]`

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

∴  (A – A^T)^T = – (A – A^T),

i.e., A – A^T = (A – A^T)^T 

∴ A – A^T is a skew-symmetric matrix.