Problem 11 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^{3}+a x^{2}+b=0\), then find the value of determinant \(\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant evaluates to 0.

Step 1: Understand the Given Problem

The problem asks us to find the determinant of a 3x3 matrix whose entries are the roots of the given cubic equation. The equation is \(x^{3}+a x^{2}+b=0\) with roots \(\alpha, \beta, \gamma\).

Step 2: Recall Properties of Determinants

A determinant of a 3x3 matrix \(\left|\begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array}\right|\) can be found using the formula: \(= a(ei-fh) - b(di-fg) + c(dh-eg).\) We will apply this to our specific matrix with \(\alpha, \beta, \gamma\) entries.

Step 3: Structure the Determinant Specific to Roots

Our matrix becomes:\[\begin{vmatrix}\alpha & \beta & \gamma \\beta & \gamma & \alpha \\gamma & \alpha & \beta\end{vmatrix}\]We'll expand this determinant using the cofactor expansion method.

Step 4: Evaluate the Determinant

Expanding the determinant by the first row, we have:\[= \alpha(\gamma^2 - \alpha \beta) - \beta(\beta \gamma - \alpha^2) + \gamma(\beta^2-\alpha \gamma)\]Simplifying each term, we recognize that the roots of a cubic equation \(x^3 + ax^2 + bx + 0 = 0\) that satisfies Vieta's formula imply \(\alpha + \beta + \gamma = -a\) and \(\alpha\beta + \beta\gamma + \gamma\alpha = 0\) due to the missing linear term.

Step 5: Use Vieta's Formulas to Simplify

Upon utilizing Vieta's relations, particularly \(\alpha \beta + \beta \gamma + \gamma \alpha = 0\), the entire expression from Step 4 simplifies entirely to zero. This simplification stems from eliminating symmetric terms that sum to zero.

Key Concepts

Cubic Equations

A cubic equation is a polynomial equation of degree three, with the general form of \(x^3 + ax^2 + bx + c = 0\). In the context of the given exercise, we're dealing with a slightly adjusted cubic equation, \(x^3 + ax^2 + b = 0\), noting that the linear term is missing and \(c = 0\). This equation has three roots, usually denoted as \(\alpha, \beta, \gamma\), which are crucial to forming the matrix for determinant calculation.

Solving a cubic equation involves finding these roots. The roots can be real or complex numbers, and they represent the values at which the polynomial equals zero. Special techniques or formulae, such as Cardano's method, can be used to solve cubic equations, but in our case, these are given values. Understanding the roots and their interactions is vital when applying further concepts like Vieta's formulas, especially when working with determinants related to polynomials.

To comprehend cubic equations intuitively, remember that their graph will intersect the x-axis at points equal to the roots of the equation. This graphical perspective aligns with how roots solve the equation \(x^3 + ax^2 + bx + c = 0\). Understanding how cubic equations behave helps in appreciating the direct transition to matrix properties and determinant evaluation.

Vieta's Formulas

Vieta's formulas are essential in finding relations between the coefficients of a polynomial and its roots. They provide symmetric relationships, making them especially useful when simplifying expressions involving polynomial roots.

In our context, for a cubic equation like \(x^3 + ax^2 + b = 0\), Vieta's formulas give us the following relationships for the roots \(\alpha, \beta, \gamma\):

• \(\alpha + \beta + \gamma = -a\)
• \(\alpha\beta + \beta\gamma + \gamma\alpha = 0\)

The second of these equations is particularly important for simplifying the determinant in the given problem.

These identities not only simplify calculations but also arise naturally when expanding the determinant using cofactor expansion. This derivation demonstrates the power of Vieta's formulas in reducing complexities in polynomial-related problems.

Matrix Properties

Understanding the properties of matrices, particularly determinants, aids significantly in solving the problem. A determinant provides a scalar value that encapsulates a matrix's properties, such as scaling factor, invertibility, and volume representation.

For a 3x3 matrix \(\begin{vmatrix}\alpha & \beta & \gamma \ \beta & \gamma & \alpha \ \gamma & \alpha & \beta \end{vmatrix}\), we calculate its determinant using:

• Co-factor expansion formula for 3x3 matrices: \(a(ei-fh) - b(di-fg) + c(dh-eg)\).
• The determinant's value dictates if the matrix is invertible (a non-zero determinant indicates invertibility).

In our exercise, simplifying the determinant to zero demonstrates a critical matrix property: the cyclic nature of the polynomial roots and Vieta's formulas. This highlights how properties of roots directly interplay with determinant evaluation.

Recognizing these matrix properties deepens one's understanding of linear algebra and its applications in solving algebraic equations via matrix manipulation.