Question

If `omega` is a non-real cube root of unity and n is not a multiple of 3, then `Delta = abs ((1, omega^n, omega^(2n)),(omega^(2n), 1, omega^n),(omega^n, omega^(2n), 1))` is equal to ____________.

Options

• 0

• None of these

• 1

• -1

Answer 1

If `omega` is a non-real cube root of unity and n is not a multiple of 3, then `Delta = abs ((1, omega^n, omega^(2n)),(omega^(2n), 1, omega^n),(omega^n, omega^(2n), 1))` is equal to 0.

Explanation:

`abs ((1, omega^n, omega^(2n)),(omega^(2n), 1, omega^n),(omega^n, omega^(2n), 1))`

Apply, `"R"_1 -> omega^n  "R"_1`

`1/omega^n abs ((omega^n, omega^(2n), omega^(3n)),(omega^(2n), 1, omega^n),(omega^n, omega^(2n), 1))` (multiply and divide by ωⁿ ) and

`omega^(3n) = 1` and taking ωⁿ common from C₁

`omega^n/omega^n abs ((1, omega^(2n), 1),(omega^n, 1, omega^n),(1, omega^(2n), 1))`

Since C₁ and C₃ are identical
So value is equal to 0