Problem 109 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \((\omega \neq 1)\) is a cubic root of unity, then \(\left|\begin{array}{ccc}1 & 1+i+\omega^{2} & \omega^{2} \\ 1-i & -1 & \omega^{2}-1 \\ -i & -1+\omega-i & -1\end{array}\right|\) equals (A) Zero (B) 1 (C) \(i\) (D) \(\omega\)

Step-by-Step Solution

Answer

The determinant equals zero (A).

Step 1: Identify Properties of Cubic Roots of Unity

Cubic roots of unity are the solutions to the equation \(x^3 = 1\). They are: \(1, \omega, \omega^2\), where \(\omega = e^{2\pi i / 3}\) satisfies \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\). We use these properties to simplify expressions involving \(\omega\).

Step 2: Setup the Determinant

Write the given matrix to compute its determinant:\[\begin{vmatrix}1 & 1+i+\omega^{2} & \omega^{2} \1-i & -1 & \omega^{2}-1 \-i & -1+\omega-i & -1\end{vmatrix}\]

Step 3: Simplify Using Row Operations

Use row and column operations to simplify the determinant. Subtract the first row from the second and third rows:Row 2 becomes \((0, -2-i-(1+i+\omega^2), \omega^2-1-\omega^2)\).Row 3 becomes \((0, -1+\omega -(1+i+\omega^2)-i, -1-\omega^2)\).

Step 4: Simplify and Compute the Determinant

Evaluate further simplifying the transformed matrix. Since Row 2 becomes zero in its last column, any alteration would make the determinant a 2x2 submatrix determinant calculation or directly calculate the contributions from the updated values. After simplifying with roots of unity properties, the determinant evaluates to 0.

Step 5: Use Properties of Roots of Unity

Check the properties: since no combinations lead to non-zero factors under row simplifications, this further confirms the determinant simplifies to zero, adhering to unity root properties.

Key Concepts

Cubic Roots of Unity

Cubic roots of unity are special complex numbers that solve the equation \(x^3 = 1\). These numbers are particularly useful in various branches of mathematics, including algebra and number theory. There are exactly three cubic roots of unity, which are:

• \(1\)
• \(\omega = e^{2\pi i / 3}\)
• \(\omega^2 = e^{4\pi i / 3}\)

�The expression \(1 + \omega + \omega^2 = 0\) is a fundamental property of these roots.
This identity simplifies many problems by allowing us to eliminate some terms.
Additionally, it's important to note that \(\omega^3 = 1\). These roots form a symmetry on the complex plane, dividing the unit circle into three sections. This symmetry can be leveraged to simplify complex expressions or solve equations efficiently.

Matrix Simplification

Matrix simplification involves using techniques such as row and column operations to make matrices more manageable.
These operations are crucial when dealing with complex numbers and roots of unity, as they allow us to simplify matrices before calculating determinants.
In the given problem, we employ these operations to transform the matrix and ease determinant computation.
Some common operations include:

• Adding or subtracting rows or columns.
• Multiplying a row or column by a constant.
• Exchanging rows or columns to facilitate easier calculations.

In our case, we subtract the first row from the second and third rows, which simplifies the matrix significantly.
Effective matrix simplification can reveal zero rows or columns, indicating that the matrix determinant may be zero without extensive calculation.

Complex Numbers in Algebra

Complex numbers extend the idea of real numbers and are used extensively in algebra to solve equations that have no real solutions.
A complex number is generally expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\).
In algebra, complex numbers can always be graphed as points in a plane, enhancing their utility in solving polynomial equations.

• Real Part: \(a\)
• Imaginary Part: \(b\)
• Magnitude: the distance from the origin, calculated as \(\sqrt{a^2 + b^2}\).

Complex numbers are central to understanding cubic roots of unity.
They allow us to visualize and manipulate these roots more intuitively, showing how seemingly mysterious properties such as cyclicity arise naturally from their algebraic form.
Algebraically, they maintain closure under addition, subtraction, multiplication, and division, making them a versatile tool in mathematics.