Question

If [a_ij]_3×3, where a_ij = 2(i – j), find A and A^T. State whether A and A^T both are symmetric or skew-symmetric matrices?

Answer 1

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

Given, a_ij = 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)]`

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

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

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