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Question
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3,5),(1,-1)]`
Solution
to suppose, A = `[(3,5),(1, -1)],` A' =`[(3,1),(5, -1)]`
So, A `1/2` (A + A') + `1/2` (A - A')
Let, P = `1/2` (A + A') = `1/2 ([(3, 5),(1, -1)]) + ([(3, 1),(5, -1)])`
`= 1/2 [(3 + 3, 5 + 1), (1 + 5, -1 -1)]`
`= 1/2 [(6, 6), (6, -2)]`
`= [(3, 3), (3, -1)]`
and, `"P'" = [(3, 3), (3, -1)] = "P",`
Therefore, the matrix P is a symmetric matrix.
Then, Q = `1/2` (A - A') = `1/2 ([(3, 5),(1, -1)]) - ([(3, 1),(5, -1)])`
`= 1/2 [(3 - 3, 5 -1), (1 - 5, -1 + 1)]`
`= 1/2 [(0,4), (-4, 0)]`
`= [(0,2), (-2, 0)]`
and, Q' = `[(0,2), (-2, 0)]` = - Q,
Hence the matrix Q is a skew symmetric matrix.
So, A = P + Q
= `[(3, 3), (3, -1)] + [(0, -2),(-2, 0)]`
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