Maharashtra State BoardHSC Commerce 12th Board Exam
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If [aij]3×3, where aij = 2(i – j), find A and AT. State whether A and AT both are symmetric or skew-symmetric matrices? - Mathematics and Statistics

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Sum

If [aij]3×3, where aij = 2(i – j), find A and AT. State whether A and AT both are symmetric or skew-symmetric matrices?

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Solution

A = `["a"_"ij"]_(3xx3) = [("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)]`

Given, aij = 2 (i – j)
∴ a11 = 2(1 – 1) = 0,  a12 = 2(1 – 2) = – 2
a13 = 2(1 – 3) = – 4,  a21 = 2(2 – 1) = 2,
a22 = 2(2 – 2) = 0,     a23 = 2(2 – 3) = – 2,
a31 = 2(3 – 1) = 4,     a32 = 2(3 – 2) = 2,
a33 = 2(3 – 3) = 0

∴ A = `[(0, -2, -4),(2, 0, -2),(4, 2, 0)]`

∴ AT = `[(0, 2, 4),(-2, 0, -2),(-4, -2, 0)]`

= `-[(0, -2, -4),(2, 0, -2),(4, 2, 0)] = - "A"`

∴ AT = – A and A = – AT 
∴ A and AT both are skew-symmetric matrices.

Concept: Algebra of Matrices
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