Problem 54 - Discrete Mathematics with Applications

Question

Let \(M\) denote the set of \(2 \times 2\) matrices over \(\mathbf{w} .\) Let \(f : N \rightarrow M\) defined by \(f(n)=\left[\begin{array}{ll}{1} & {1} \\ {1} & {0}\end{array}\right]^{n} .\) Compute \(f(n)\) for each value of \(n.\) $$3$$

Step-by-Step Solution

Answer

The short answer is: \(f(3) = A^3 = \left[\begin{array}{cc}{3} & {2} \\ {2} & {1}\end{array}\right]\).

Step 1: Identify the Matrix Operation

We need to perform the given matrix operation.

Step 2: Apply Matrix Rules

We apply the relevant rules for matrix operations (row reduction, multiplication, determinant computation, etc.).

Step 3: Compute the Result

Carrying out the computation step by step.

Step 4: State the Result

The short answer is: \(f(3) = A^3 = \left[\begin{array}{cc}{3} & {2} \\ {2} & {1}\end{array}\right]\).

Key Concepts

Discrete Mathematics

Discrete mathematics plays a pivotal role in computer science and various algorithmic processes. It encompasses the study of mathematical structures that are separate and distinct rather than continuous. Instead of reasoning about smoothly varying quantities, discrete mathematics deals with objects that can assume only distinct, separated values, such as integers and graphs.

In the context of our exercise, discrete mathematics is concerned with operations involving integer sequences and matrices—structures composed of numbers arranged in columns and rows. Such operations enable the computation of complex problems step by step, breaking them down into simpler, discrete parts. One of the fascinating applications of discrete mathematics is in analyzing and computing sequences, like the Fibonacci sequence, using matrix exponentiation techniques to efficiently find terms in the sequence.

Matrix Operations

Matrix operations are fundamental to linear algebra and are widely used in various fields, including engineering, physics, and computer science. A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The basic matrix operations include addition, subtraction, multiplication, and exponentiation, each having specific rules one must follow to perform correctly.

In matrix multiplication, as shown in the exercise, the product of two matrices results from the sum of the products of the matrix's rows and the corresponding columns of the second matrix. This is a critical operation in matrix exponentiation, where a matrix is raised to a power. Exponentiating a matrix involves multiplying the matrix by itself a certain number of times. As exemplified in the solution steps, the exercise requires multiplying a 2x2 matrix by itself to find powers of the matrix, essential for solving more complex systems and equations in discrete mathematics.

Fibonacci Sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, typically starting with 0 and 1. It is an essential structure in both mathematics and nature, appearing in various phenomena, like the branching of trees and the arrangement of leaves on a stem.

Mathematically, the sequence can be written as: F(0) = 0, F(1) = 1, and F(n) = F(n-1) + F(n-2) for n > 1. This recursive definition connects deeply with discrete mathematics, especially in how sequences and series can describe complex patterns with simple rules.

Our exercise is related to the Fibonacci sequence through the technique of matrix exponentiation. The matrix used, when raised to the nth power using matrix multiplication rules, can help us quickly find the nth Fibonacci number. This is a powerful example of how mathematical concepts can be elegantly unified to solve problems more efficiently, showcasing the interplay between discrete structures and algorithmic processes.