Complex Numbers
Cube Roots of Unity
Grade 11
Question:
<p>Let ω be the complex number \(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\), where <i>i</i> = √−1, then the number of distinct complex numbers <i>z</i> satisfying</p><p>\[\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 0\]</p><p>is equal to</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: The determinant of a matrix with cube roots of unity has special properties that yield a cubic equation with three distinct roots.
<p>The determinant equation represents a cubic equation in <i>z</i>. Since ω is a primitive cube root of unity satisfying ω³ = 1 and 1 + ω + ω² = 0, the structure of the determinant yields exactly 3 distinct solutions.</p>
Correct Answer: D