Find The Absolute Difference Between The Largest And Smallest Eigenvalue Of A 3x3 Matrix Whose Eigenvalues Form An Arithmetic Progression, Given The Characteristic Equation D^3 - 12d^2 + 39d - 28 = 0.

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In the realm of linear algebra, eigenvalues and eigenvectors hold immense significance. They offer profound insights into the behavior of linear transformations and matrices. Understanding these concepts is crucial in various fields such as physics, engineering, computer science, and economics. Before diving into the specific problem, let's establish a solid foundation by exploring the fundamental definitions and properties of eigenvalues and characteristic equations.

An eigenvector of a square matrix A is a non-zero vector that, when multiplied by A, results in a scalar multiple of itself. This scalar is known as the eigenvalue, often denoted by λ (lambda). Mathematically, this relationship is expressed as:

Av = λ*v

where:

  • A is the square matrix.
  • v is the eigenvector.
  • λ is the eigenvalue.

To find the eigenvalues of a matrix, we solve the characteristic equation. The characteristic equation is derived from the equation:

(A - λI)*v = 0

where I is the identity matrix of the same size as A. For a non-trivial solution (i.e., v ≠ 0), the determinant of (A - λI) must be zero:

det(A - λI) = 0

This determinant results in a polynomial equation in λ, known as the characteristic polynomial. The roots of this polynomial are the eigenvalues of the matrix A. For a 3x3 matrix, the characteristic polynomial will be a cubic equation, meaning there will be three eigenvalues (counting multiplicities).

The problem presents a 3x3 matrix, which we'll refer to as A_{3x3}. We are given that its characteristic equation is:

d^3 - 12d^2 + 39d - 28 = 0

where d represents the eigenvalues. The crucial piece of information is that the eigenvalues are in arithmetic progression (A.P.). This provides us with a critical constraint that simplifies the problem-solving process. Our goal is to find the absolute difference between the largest and smallest eigenvalue.

To solve this, we'll use the following steps:

  1. Represent the eigenvalues in A.P.: Since the eigenvalues are in arithmetic progression, we can represent them as a - b, a, and a + b, where a is the middle term and b is the common difference.
  2. Relate eigenvalues to the characteristic equation: The coefficients of the characteristic equation are related to the sums and products of the eigenvalues. We'll use Vieta's formulas to establish these relationships.
  3. Solve for the eigenvalues: Using the relationships derived from Vieta's formulas and the A.P. property, we'll solve for the values of a and b, thereby finding the eigenvalues.
  4. Calculate the absolute difference: Finally, we'll calculate the absolute difference between the largest (a + b) and smallest (a - b) eigenvalues.

Vieta's formulas provide a powerful connection between the coefficients of a polynomial and its roots. For a cubic equation of the form:

x^3 + px^2 + qx + r = 0

with roots x₁, x₂, and x₃, Vieta's formulas state:

  • x₁ + x₂ + x₃ = -p
  • x₁x₂ + x₁x₃ + x₂x₃ = q
  • x₁x₂x₃ = -r

In our case, the characteristic equation is:

d^3 - 12d^2 + 39d - 28 = 0

So, p = -12, q = 39, and r = -28. Let the eigenvalues be d₁ = a - b, d₂ = a, and d₃ = a + b. Applying Vieta's formulas:

  1. Sum of eigenvalues:

    (a - b) + a + (a + b) = -(-12)

    3a = 12

    a = 4

  2. Sum of pairwise products:

    (a - b)a + (a - b)(a + b) + a(a + b) = 39

    a² - ab + a² - b² + a² + ab = 39

    3a² - b² = 39

  3. Product of eigenvalues:

    (a - b)(a)(a + b) = -(-28)

    a(a² - b²) = 28

We already found that a = 4. Substituting this into the second equation derived from Vieta's formulas:

3(4²) - b² = 39

48 - b² = 39

b² = 9

b = ±3

Now that we have the values of a and b, we can determine the eigenvalues. We have two possible sets of eigenvalues depending on whether b is 3 or -3. However, the absolute difference between the largest and smallest eigenvalue will be the same in both cases. Let's consider b = 3:

  • d₁ = a - b = 4 - 3 = 1
  • d₂ = a = 4
  • d₃ = a + b = 4 + 3 = 7

The eigenvalues are 1, 4, and 7, which are indeed in arithmetic progression.

The largest eigenvalue is 7, and the smallest eigenvalue is 1. Therefore, the absolute difference between the largest and smallest eigenvalue is:

|7 - 1| = 6

If we consider b = -3, the eigenvalues would be 7, 4, and 1, which are the same eigenvalues in reverse order. The absolute difference would still be |1 - 7| = 6.

Therefore, the absolute difference between the largest and smallest eigenvalue is 6. This corresponds to option (d) in the given problem.

This problem highlights the powerful interplay between the characteristic equation, eigenvalues, and the properties of arithmetic progressions. The key takeaways from this problem are:

  • Understanding the definition of eigenvalues and how to find them using the characteristic equation.
  • Applying Vieta's formulas to relate the coefficients of a polynomial to its roots.
  • Utilizing the given constraints, such as the arithmetic progression property, to simplify the problem and solve for the unknowns.
  • Recognizing that eigenvalues provide critical information about the behavior of a matrix or linear transformation.

By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of eigenvalue problems in linear algebra and related fields. The ability to connect seemingly disparate mathematical concepts, like Vieta's formulas and arithmetic progressions, is a hallmark of effective problem-solving in mathematics. Remember to always look for constraints and relationships within the problem statement, as they often provide the key to unlocking the solution.