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प्रश्न
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(1, 5),(-1, 2)]`
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उत्तर
Let, A = `[(1, 5),(-1, 2)]`
⇒ `A' = [(1, -1),(5, 2)]`
`A + A' = [(1, 5),(-1, 2)] + [(1, -1),(5, 2)]`
= `[(1 + 1, 5 - 1),(-1 + 5, 2 + 2)]`
= `[(2, 4),(4, 4)]`
∴ `1/2 (A + A') = 1/2 [(2, 4),(4, 4)]`
= `[(1, 2),(2, 2)]`
And A – A' = `[(1, 5),(-1, 2)] - [(1, -1),(5, 2)]`
= `[(1 - 1, 5 + 1),(-1 - 5, 2 - 2)]`
= `[(0, 6),(-6, 0)]`
∴ `1/2 (A - A') = 1/2 [(0, 6),(-6, 0)] = [(0, 3),(-3, 0)]`
`A = 1/2 (A + A') + 1/2 (A - A')`
= `[(1, 2),(2, 2)] + [(0, 3),(-3, 0)]` = A
Symmetric matrices + Skew symmetric matrices
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