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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]3 × 3 = `[("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)]`
Given that: 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)]`
∵ aij = − aij for all i and j
∵ A is a skew-symmetric matrix.
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