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Question
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(1,5),(-1,2)]`
Solution
Let, A = `[(1,5),(-1,2)]`
`=> A' = [(1,-1),(5,2)]`
`A + A' = [(1,5),(-1,2)] + [(1,-1),(5,2)]`
`= [(1 + 1, 5 - 1),(-1 + 5, 2 + 2)]`
`= [(2,4),(4,4)]`
`therefore 1/2 (A + A') = 1/2 [(2,4),(4,4)]`
`= [(1,2),(2,2)]`
and, A - A' = `[(1,5),(-1,2)] - [(1,-1),(5,2)]`
`= [(1 - 1, 5 + 1),(-1 -5, 2 - 2)]`
`= [(0,6),(-6,0)]`
`therefore 1/2 (A - A') = 1/2 [(0,6),(-6,0)] = [(0,3),(-3,0)]`
`A = 1/2 (A + A') + 1/2 (A - A')`
`= [(1,2),(2,2)] + [(0,3),(-3,0)] = A`
Symmetric matrices + Skew symmetric matrices
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