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Question
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3, 5),(1, -1)]`
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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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