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Question
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
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Solution
A = `[(3,3, -1),(-2, -2, 1),(-4, -5, 2)]`
⇒ A' = `[(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`
∴ A + A' = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)] + [(3, -2, -4),(3, -2, -5),(-1, -1, 2)]`
= `[(3 + 3, 3 - 2, -1 - 4),(-2 + 3, -2 - 2, 1 -5),(-4 - 1, -5 + 1, 2 + 2)]`
= `[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`
∴ `1/2 (A + A') = 1/2 [(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`
= `[(3, 1/2, -5/2),(1/2, -2, -2),(-5/2, -2, 2)]`
And A – A' = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)] - [(3, -2, -4),(3, -2, -5),(-1, -1, 2)]`
= `[(3 - 3, 3 + 2, -1 + 4),(-2 - 3, -2 + 2, 1 + 5),(-4 + 1, -5 - 1, 2 - 2)]`
= `[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`
`1/2 (A - A') = 1/2 [(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`
= `[(0, 5/2, 3/2),(-5/2, 0, 3),(-3/2, -3, 0)]`
`A = 1/2 (A + A') + 1/2 (A - A')`
= `[(3, 1/2, -5/2),(1/2, -2, -2),(-5/2, -2, 2)] + [(0, 5/2, 3/2),(-5/2, 0, 3),(-3/2, -3, 0)]`
= `[(3 + 0, 1/2 + 5/2, -5/2 + 3/2),(1/2 - 5/2, -2 + 0, -2 + 3),(-5/2 - 3/2, -2 -3, 2 + 0)]`
= `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]` = A
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