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Question
Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where A = `[(2, 4, -6),(7, 3, 5),(1, -2, 4)]`
Solution
We have A = `[(2, 4, -6),(7, 3, 5),(1, -2, 4)]`
Then A' = `[(2, 7, 1),(4, 3, -2),(-6, 5, 4)]`
Hence `("A" + "A'")/2 = 1/2 [(4, 11, -5),(11, 6, 3),(-5, 3, 8)]`
= `[(2, 11/2, (-5)/2),(11/2, 3, 3/2),((-5)/2, 3/2, 4)]`
and `("A" - "A'")/2 = 1/2 [(0, -3, -7),(3, 0, 7/2),(7, -7, 0)]`
= `[(0, (-3)/2, (-7)/2),(3/2, 0, 7/2),(7/2, (-7)/2, 0)]`
Therefore,
`("A" + "A'")/2 + ("A" - "A'")/2 = [(2, 11/2, (-5)/2),(11/2, 3, 3/2),((-5)/2, 3/2, 4)] + [(0, (-3)/2, (-7)/2),(3/2, 0, 7/2),(7/2, (-7)/2, 0)]`
= `[(2, 4, -6),(7, 3,5),(1,-2, 4)]`
= A
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