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