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Question
Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.
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Solution
We have, A = `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]`
We know that A = `("A" + "A'")/2 + ("A" - "A'")/2`
Where `("A" + "A'")/2` is symmetric and `("A" - "A'")/2` is skew-symmetric
∴ A' = `[(2, 1, 4),(3, -1, 1),(1, 2, 2)]`
Now, `("A" + "A'")/2 = ([(2, 3, 1),(1, -1, 2),(4, 1, 2)] + [(2, 1, 4),(3, -1, 1),(1, 2, 2)])/2`
= `1/2 [(4, 4, 5),(4, -2, 3),(5, 3, 4)]`
= `[(2, 2, 5/2),(2, -1, 3/2),(5/2, 3/2, 2)]`
And `("A" - "A'")/2 = ([(2, 3, 1),(1, -1, 2),(4, 1, 2)] - [(2, 1, 4),(3, -1, 1),(1, 2, 2)])/2`
= `1/2 [(0, 2, -3),(-2, 0, 1),(3, -1, 0)]`
= `[(0,1, (-3)/2),(-1, 0, 1/2),(3/2, (-1)/2, 0)]`
∴ A = `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]`
= `[(2, 2, 5/2),(2, -1, 3/2),(5/2, 3/2, 2)] + [(0, 1, (-3)/2),(-1, 0, 1/2),(3/2, 1/2, 0)]`
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