Question

Prove that A + A^T is a symmetric and A – A^T is a skew symmetric matrix, where A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`

Answer 1

A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`

∴ A^T = `[(1, 3, -2),(2, 2, -3),(4, 1, 2)]`

∴ A + A^T = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)] + [(1, 3, -2),(2, 2, -3),(4, 1, 2)]`

= `[(1 + 1, 2 + 3, 4 - 2),(3 + 2, 2 + 2, 1 - 3),(-2 + 4, -3 + 1, 2 + 2)]`

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

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

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

A – A^T = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)] - [(1, 3, -2),(2, 2, -3),(4, 1, 2)]`

= `[(1 - 1, 2 - 3, 4 + 2),(3 - 2, 2 - 2, 1 + 3),(-2 - 4, -3 - 1, 2 - 2)]`

∴ A – A^T = `[(0, -1, 6),(1, 0, 4),(-6, -4, 0)]`

∴ (A – A^T)^T = `[(0, 1, -6),(-1, 0, -4),(6, 4, 0)]`

= `-[(0, -1, 6),(1, 0, 4),(-6, -4, 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.