Question

Construct the matrix A = [a_ij]_3×3, where a_ij = 1 – j. State whether A is symmetric or skew–symmetric

Answer 1

A = [a_ij]_3×3

Where a_ij = 1 – j

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

a_ij = i – j

a₁₁ = 1 – 1 = 1

a₁₂ = 1 – 2 = – 1

a₁₃ = 1 – 3 = – 2

a₂₁ = 2 – 1 = 1

a₂₂ = 2 – 2 = 1

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)]`

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

A^T = – A

∴ A is skew-symmetric.