Question

Construct the matrix A = [a_ij]_3×3 where a_ij = i − j. State whether A is symmetric or skew-symmetric.

Answer 1

A = [a_ij]_3x3  

∴ A = `[("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)]`

Given, a_ij = 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)]`

∴ A^T = `[(0, 1, 2), (-1, 0, 1),(-2, -1, 0)]`

= `-[(0, -1, -2),(1, 0, -1),(2, 1, 0)]`

∴ A^T = – A i.e., A = – A^T
∴ A is a skew-symmetric matrix.