If n is not a multiple of 3, the value of the determinant

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If $n$ is not a multiple of 3, the value of the determinant $\left|\begin{array}{ccc}\omega^{2 n} & 1 & \omega^{n} \\ 1 & \omega^{n} & \omega^{2 n} \\ \omega^{n} & \omega^{2 n} & 1\end{array}\right|$, where $\omega$ is a complex cube root of unity is: (a) $\omega$ (b) $\omega^{2}$ (c) 1 (d) 0

Key Concepts

Determinant of a Matrix

A determinant is a scalar value that is computed from the elements of a square matrix and provides important information about the matrix, such as whether it is invertible. Key properties of determinants include the fact that if the rows or columns of a matrix are linearly dependent, the determinant becomes zero. In problems involving structured matrices, the interplay between the elements can lead to such dependencies, which directly causes the determinant to vanish.

Cube Roots of Unity

Cube roots of unity are the solutions to the equation x³ = 1 in the complex number system. Besides the trivial solution x = 1, the two non-real cube roots, commonly denoted ? and ?², have the crucial property that 1 + ? + ?² = 0. This relation is invaluable for simplifying expressions in problems where these roots appear, as it allows one to express sums and products in a more tractable form.

Modular Arithmetic in Exponents

Modular arithmetic is used to simplify expressions involving powers, especially when dealing with periodic functions such as those arising from roots of unity. Since the cube roots of unity satisfy ?³ = 1, any exponent can be reduced modulo 3. This reduction is particularly useful when the exponent (or a multiple of it) in an expression, such as ??, is involved, thereby streamlining the algebra involved in computing determinants or simplifying expressions.

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