Question

If \(\mathrm{w}\) is one of the cube root of 1 other then 1 then $$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & -1-\mathrm{w}^{2} & \mathrm{w}^{2} \\ 1 & \mathrm{w}^{2} & \mathrm{w}^{4} \end{array}\right|= $$ (a) \(3 \mathrm{w}\) (b) \(3 \mathrm{w}(\mathrm{w}-1)\) (c) \(3 \mathrm{w}^{2}\) (d) \(3 \mathrm{w}(1-\mathrm{w})\)

Short answer

(d) \(3w(1-w)\)

Step-by-step solution

(Step 1: Understand the given cube root of unity)

The cube roots of unity are solutions to the equation \(x^3 =1\). These roots are \(1, w, w^2\), where \(w\) and \(w^2\) are complex roots. Also, note that \(1, w, w^2\) are all distinct and \(w ≠ 1\) and \(w^3 = 1\).

(Step 2: Express \(w^{2}\) and \(w^{4}\) in terms of \(w\) and \(1\))

Given that \(w^3 = 1\), we could also infer that \(w^{2} = w^{-1}\) and \(w^{4} = w^{-2}\) because the indices are the same in modulo 3. Since \(w^{-1}\) is the multiplicative inverse of \(w\), and \(w^{-2}\) = \((w^{-1})^2\), it simplifies to \(w^{2} = w^{-1} = 1/w\) and \(w^{4} = 1/w^2\).

(Step 3: Substitute the expressions from Step 2 back to the determinant)

Replace \(w^2\) with \(1/w\) and \(w^4\) with \(1/w^2\) in the matrix. The determinant will, therefore, become \[ \begin{vmatrix} 1 & 1 & 1 \\ 1 & -1-1/w & 1/w \\ 1 & 1/w & 1/w^2 \end{vmatrix}. \]

(Step 4: Apply Sarrus' rule to compute the determinant)

According to Sarrus' rule, the determinant of the matrix is calculated as follows: \[ (1 \cdot (-1-1/w) \cdot 1/w^2) + (1 \cdot 1/w \cdot 1) + (1 \cdot 1 \cdot 1/w) - ((1 \cdot 1/w \cdot 1/w^2) + (1 \cdot (-1-1/w) \cdot 1) + (1 \cdot 1 \cdot (-1 - 1/w))). \] When you simplify this equation, you will get, \(-1/w^2 -1/w +3-1/w-1/w^2 -1\) or \(3w - 2 - 2/w -1/w^2\) or \(3w(1-w)\) which is the option (d). Therefore, the answer is: (d) \(3w(1-w)\).

Key concepts

Determinants

A determinant is a special number that can be calculated from a square matrix. This value is pivotal in solving systems of linear equations, among other applications in linear algebra. To calculate the determinant of a 3x3 matrix, like the one given in the original exercise, you can use the Sarrus' rule. This method involves multiplying across diagonals in both directions, adding the results from one direction, and then subtracting the results from the other.

• First, you multiply the items on the main diagonals from left to right, which consists of the elements from the top left to the bottom right.
• Then, you multiply the items on the diagonals running from the top right to the bottom left.
• You add the results from the first set of products and subtract the values from the second set from it.

The resulting sum gives you the determinant. In this matrix's case, the expressions involve complex components, requiring an understanding of complex numbers for calculation.

Complex Numbers

Complex numbers expand the idea of one-dimensional real numbers to a two-dimensional space. This is achieved through the combination of a real number and an imaginary number, typically noted as a + bi, where i is the square root of -1.

For cube roots of unity, we deal with particular complex numbers which, when cubed, equal one. The roots are 1, \( w \), and \( w^2 \), forming an equilateral triangle in the complex plane centered at the origin. Some important properties include:

• \( w^3 = 1 \)
• \( 1 + w + w^2 = 0 \)

These properties allow simplification of expressions, as seen in the exercise, where the cube roots of unity's unique attributes are vital to simplifying the determinant calculation.

Matrix Algebra

Matrix algebra allows for the handling and manipulation of complex arrays of numbers, expanding the capabilities of solving mathematical problems. In matrix algebra, each position in a matrix is carefully used, and operations such as addition, subcontraction, scalar multiplications, and determinant calculations are performed.

For example, understanding the way matrix columns and rows interact is crucial, as performed in calculating the determinant.

• First, ensure that the elements are correctly placed within the matrix.
• Perform operations systematically, respecting the matrix's dimensions and properties of numbers involved, such as complex numbers.

The interaction of elements within a matrix, governed by the rules of matrix algebra, provides solutions to systems of equations and guides the determinant's computation in complex scenarios.