Question

For what value of x, the matrix A = `[(0, 1, -2),(-1, 0, x^3),(2, -3, 0)]` is skew – symmetric

Answer 1

A = `[(0, 1, -2),(-1, 0, x^3),(2, -3, 0)]`

A^T = `[(0, -1, 2),(1, 0, -3),(-2, x^3, 0)]`

The matrix A is skew-symmetric if A = – A^T

`[(0, 1, -2),(-1, 0, x^3),(2, -3, 0)] = - [(0, -1, 2),(1, 0, -3),(-2, x^3, 0)]`

`[(0, 1, -2),(-1, 0, x^3),(2, -3, 0)] + [(0, -1, 2),(1, 0, -3),(-2, x^3, 0)] = [(0, 0, 0),(0, 0, 0),(0, 0, 0)]`

`[(0 + 0, 1 - 1, -2 + 2),(-1 + 1, 0 + 0, x^3 - 3),(2 - 2, -3 x^3, 0 + 0)] = [(0, 0, 0),(0, 0, 0),(0, 0, 0)]`

`[(0, 0, 0),(0, 0, x^3 - 3),(0, x^3 - 3, 0)] = [(0, 0, 0),(0, 0, 0),(0, 0, 0)]`

Equating the corresponding entries

x³ – 3 = 0

x³ = 3

⇒ x = `3^(1/3)`