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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)]`
`therefore "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)]`
`therefore 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.
`therefore "and" ("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)]`
`therefore 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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