Problem 6 - Applied Discrete Structures

Question

Recall that \(p(x)=x^{2}-5 x+6\) is called a polynomial, or more specifically, a polynomial over \(\mathbb{R},\) where the coefficients are elements of \(\mathbb{R}\) and \(x \in \mathbb{R}\). Also, think of the method of solving, and solutions of, \(x^{2}-5 x+6=0\). We would like to define the analogous situation for \(2 \times 2\) matrices. First define where \(A\) is a \(2 \times 2\) matrix \(p(A)=A^{2}-5 A+6 I\). Discuss the method of solving and the solutions of \(A^{2}-5 A+6 I=0\).

Step-by-Step Solution

Answer

Solutions are matrices \( A = 2I \) and \( A = 3I \).

Step 1: Understanding the Problem

We are given a matrix polynomial equation \( A^2 - 5A + 6I = 0 \), where \( A \) is a \( 2 \times 2 \) matrix and \( I \) is the \( 2 \times 2 \) identity matrix. Our goal is to find matrices \( A \) that satisfy this equation.

Step 2: Review of Polynomial Solving Techniques

For scalar polynomials such as \( x^2 - 5x + 6 = 0 \), we find the roots by factoring or using the quadratic formula. The equation factors to \( (x-2)(x-3) = 0 \), giving roots \( x = 2 \) and \( x = 3 \). These roots inspire us to consider how the roots relate to matrix solutions.

Step 3: Matrix Polynomial Factorization

The scalar factorization \( (x-2)(x-3) = 0 \) suggests for matrices, we can similarly factor the matrix polynomial: \( (A-2I)(A-3I) = 0 \). This means \( A \) must either satisfy \( A - 2I = 0 \) or \( A - 3I = 0 \), or both.

Step 4: Solving for Matrix Solutions

Solving \( A - 2I = 0 \) implies \( A = 2I \) (i.e., \( A \) is a scalar matrix with eigenvalue \(2\)). Solving \( A - 3I = 0 \) implies \( A = 3I \) (i.e., \( A \) is a scalar matrix with eigenvalue \(3\)). Hence, solutions to the original matrix equation include these forms.

Step 5: Verification of Solutions

To verify, substitute \( A = 2I \) and \( A = 3I \) into the original equation. For \( A = 2I \), we calculate \( (2I)^2 - 5(2I) + 6I = 4I - 10I + 6I = 0 \). Similarly, for \( A = 3I \), \( (3I)^2 - 5(3I) + 6I = 9I - 15I + 6I = 0 \), confirming both satisfy the equation.

Step 6: Conclusion: Results

The solutions to the matrix equation \( A^2 - 5A + 6I = 0 \) are scalar matrices \( A = 2I \) and \( A = 3I \). This shows that eigenvalues of the matrix solutions are analogous to the roots of the scalar polynomial.

Key Concepts

Eigenvalues of Matrix Polynomials

In mathematics, understanding eigenvalues is crucial when working with matrices, especially polynomial matrices. Eigenvalues are special numbers associated with a matrix that reveal significant properties about the matrix's structure and operations.
For any square matrix, the eigenvalues can be found by solving the characteristic equation, which typically looks like \( \det(A - \lambda I) = 0 \). Here, \( A \) is the matrix, \( \lambda \) represents the eigenvalues, and \( I \) is the identity matrix. Simply put, eigenvalues are the roots of this equation.
In the context of the provided exercise, we see that the eigenvalues of the matrices \( A \) are those numbers which make the scalar polynomial roots of the characteristic equation. When we say \( A \) satisfies the polynomial \( A^2 - 5A + 6I \), it implies that the matrix can be broken down or factored similar to how one factors polynomials, helping us find the eigenvalues. In this case, eigenvalues are akin to the number 2 and 3, which correspond to the solutions of the scalar equation \( x^2 - 5x + 6 = 0 \).
Understanding eigenvalues is vital as they help to simplify complex matrix operations, characterizing transformations and providing insights into the matrix's behavior.

2x2 Matrices in Matrix Polynomials

2x2 matrices play a fundamental role in linear algebra, often appearing in simplified versions of more complex mathematical problems due to their computational manageability.
This particular exercise focuses on solving a matrix polynomial where \( A \) is a 2x2 matrix. The expression involves squaring the matrix, scaling it, and adding a scaled identity matrix. Here, the identity matrix \( I \) of order 2 acts as the multiplicative identity, much like the number 1 in ordinary arithmetic.2x2 matrices are used for their convenience in this exercise, as they allow deeper understanding without overwhelming complexity. Operations involving these matrices form the backbone for solving the matrix polynomial equation \( A^2 - 5A + 6I = 0 \).

Benefits of working with 2x2 matrices include the ability to:

• Easily perform matrix multiplication and addition.
• Directly visualize transformations and properties.
• Investigate fundamental concepts like determinants and inverses in a compact form.

2x2 matrices thus provide a foundational tool in mathematical education and practical applications, equipping students with the necessary skills for advanced studies.

Matrix Factorization in Solving Polynomials

Matrix factorization, an essential technique in mathematics, involves decomposing matrices into simpler, product components which simplify calculations.
In solving the matrix polynomial equation \( A^2 - 5A + 6I = 0 \), matrix factorization is used to break down the polynomial into simpler forms, analogous to factoring a quadratic equation. This can mean writing the matrix polynomial as \( (A - 2I)(A - 3I) = 0 \).
By employing this method, we learn that there's a direct analogy between factoring scalar polynomials and matrix polynomials. The resultant factors \( A - 2I \) or \( A - 3I \) indicate solutions being \( A = 2I \) and \( A = 3I \), respectively, pinning them as scalar matrices equivalent to the eigenvalues of 2 and 3 from the polynomial roots.
Benefits of matrix factorization include:

• Ease of computation, providing streamlined calculations.
• Ability to identify specific matrix forms that satisfy given conditions.
• Insight into structural properties and potential simplifications.

Through matrix factorization, students gain a powerful tool for tackling complex mathematical problems, enhancing their analytical and problem-solving abilities.