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Question
Construct the matrix A = [aij]3×3 where aij = i − j. State whether A is symmetric or skew-symmetric.
Solution
A = [aij]3x3
∴ A = `[("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)]`
Given, aij = i – j
∴ a11 = 1 – 1 = 0, a12 = 1 – 2 = – 1, a13 = 1 – 3 = – 2
a21 = 2 – 1= 1, a22 = 2 – 2 = 0, a23 = 2 – 3 = – 1,
a31 = 3 – 1 = 2, a32 = 3 - 2 = 1, a33 = 3 – 3 = 0
∴ A = `[(0, -1, -2), (1, 0, -1),(2, 1, 0)]`
∴ AT = `[(0, 1, 2), (-1, 0, 1),(-2, -1, 0)]`
= `-[(0, -1, -2),(1, 0, -1),(2, 1, 0)]`
∴ AT = – A i.e., A = – AT
∴ A is a skew-symmetric matrix.
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