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प्रश्न
Express the following matrix as the sum of a symmetric and a skew symmetric matrix
`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
उत्तर
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 = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
∴ AT = `[(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`
∴ A + AT = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)] + [(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`
= `[(6, 1, -5),(1, -4, -4),(-5, -4, 4)]`
Also, A – AT = `[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)] - [(3, -2, -4),(3, -2, -5),(-1, 1, 2)]`
= `[(0, 5, 3),(-5, 0, 6),(-3, -6, 0)]`
Let P = `1/2("A" + "A"^"T") = [(3, 1/2, (-5)/2),(1/2, -2, -2),((-5)/2, -2, 2)]`
and Q = `1/2("A" - "A"^"T") = [(0, 5/2, 3/2),((-5)/2, 0, 3),((-3)/2, -3, 0)]`
∴ P is a symmetric matrix ...[∵ aij = aij]
and Q is a skew symmetric matrix
∴ A = P + Q ...[∵ aij = – aij]
∴ 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)]`
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