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Question
Express each of the following matrix as the sum of a symmetric and a skew symmetric matrix `[(4, -2),(3, -5)]`.
Solution
A square matrix A can be expressed as the sum of a symmetric and a skew-symmetric matrix as
A = `(1)/(2)("A" + "A"^"T") + (1)/(2)("A" - "A"^"T")`
Let A = `[(4, -2),(3, -5)]`
∴ AT = `[(4, 3),(-2, -5)]`
∴ A + AT = `[(4, -2),(3, -5)] + [(4, 3),(-2, -5)]`
= `[(4 + 4, -2 + 3),(3 - 2, -5 - 5)]`
= `[(8, 1),(1, -10)]`
Also, A – AT = `[(4, -2),(3, -5)] - [(4, 3),(-2, -5)]`
= `[(4 - 4, -2 - 3),(3 + 2, -5 + 5)]`
= `[(0, -5),(5, 0)]`
Let P = `(1)/(2)("A" + "A"^"T")`
= `(1)/(2)[(8, 1),(1, -10)]`
= `[(4, 1/2),(1/2, -5)]`
and
Q = `(1)/(2)("A" - "A"^"T")`
= `(1)/(2)[(0, -5),(5, 0)]`
= `[(0, -(5)/(2)),(5/2, 0)]`
∴ P is a symmetric matrix ...[∵ aij = aij]
and Q is a skew-symmetric matrix. ...[∵ aij = – aij]
∴ A = P + Q
∴ A = `[(4, 1/2),(1/2, -5)] + [(0, -(5)/(2)),(5/2, 0)]`.
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