Question

If ω is a complex cube root of unity and A = `[(ω,0,0),(0,ω^2,0),(0,0,1)]` then A^-1 = ?

Options

• `[(ω^2,0,0),(0,ω,0),(0,0,1)]`

• `[(1,0,0),(0,1,0),(0,0,1)]`

• `[(1,0,0),(0,ω^2,0),(0,0,ω)]`

• `[(0,0,ω),(0,ω^2,0),(1,0,0)]`

Answer 1

`[(ω^2,0,0),(0,ω,0),(0,0,1)]`

Explanation:

Given, ω is a complex cube root of unity

∴ ω³ = 1

Given matrix A can be written as A = IA

`=> [(ω,0,0),(0,ω^2,0),(0,0,1)] = [(1,0,0),(0,1,0),(0,0,1)]"A"`

On apply `"R"_1 -> omega^2 "R"_1, "R"_2 => omega "R"_2,` we get

`=> [(ω^3,0,0),(0,ω^3,0),(0,0,1)] = [(ω^2,0,0),(0,ω,0),(0,0,1)]`A

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

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