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प्रश्न
Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`
उत्तर
A = `[(1, 2, 4),(3, 2, 1),(-2, -3, 2)]`
∴ AT = `[(1, 3, -2),(2, 2, -3),(4, 1, 2)]`
∴ A + AT = `[(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 + AT = `[(2, 5, 2),(5, 4, -2),(2, -2, 4)]`
∴ (A + AT)T = `[(2, 5, 2),(5, 4, -2),(2, -2, 4)]`
∴ (A + AT)T = A + AT i.e., A + AT = (A + AT)T
∴ A + AT is a symmetric matrix.
A – AT = `[(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 – AT = `[(0, -1, 6),(1, 0, 4),(-6, -4, 0)]`
∴ (A – AT)T = `[(0, 1, -6),(-1, 0, -4),(6, 4, 0)]`
= `-[(0, -1, 6),(1, 0, 4),(-6, -4, 0)]`
∴ (A – AT)T = – (A – AT)
i.e., A – AT = – (A – AT)T
∴ A – AT is a skew symmetric matrix.
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