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Question
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
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Solution
A = `[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
⇒ A = `[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
∴ A + A' = `[(6,-2,2),(-2,3,-1),(2,-1,3)] + [(6,-2,2),(-2,3,-1),(2,-1,3)]`
= `[(6 + 6, -2 - 2, 2 + 2),(-1 - 1, 3 + 3, -1 - 1),(2 + 2, -1 - 1, 3 + 3)]`
= `[(12, -4, 4),(-4, 6, -2),(4, -2, 6)]`
∴ `1/2 (A + A') = 1/2 [(12, -4, 4),(-4, 6, -2),(4, -2, 6)]`
= `[(6, -2, 2),(-2, 3, -1),(4, -1, 3)]` is a symmetric matrix.
∴ (A – A) = `[(6, -2, 2),(-2, 3, -1),(4, -1, 3)] - [(6, -2, 2),(-2, 3, -1),(4, -1, 3)]`
= `[(0, 0, 0),(0, 0, 0),(0, 0, 0)]`
∴ `1/2 (A - A') + 1/2 [(0, 0, 0),(0, 0, 0),(0, 0, 0)] = 0`
Hence, `A = 1/2 (A + A') + 1/2 (A - A')`
= `[(6, -2, 2),(-2, 3, -1),(2, -1, 3)] + [(0, 0, 0),(0, 0, 0),(0, 0, 0)]`
= `[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]` = A
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