Problem 55 - 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 \(n.\) $$4$$

Step-by-Step Solution

Answer

In conclusion, for \(n=4\), the function \(f(n)\) is equal to the matrix \(f(4) =\left[\begin{array}{ll}{5} & {3} \\\ {3} & {2}\end{array}\right]\).

Step 1: Understand the given matrix

We're given the matrix \(M = \left[\begin{array}{ll}{1} & {1} \\\ {1} & {0}\end{array}\right]\). This is a 2x2 matrix containing four elements: \(M_{11} = 1\), \(M_{12} = 1\), \(M_{21} = 1\), and \(M_{22} = 0\). Now, let's try to find the nth power of M.

Step 2: Find a general formula for M^n

Firstly, compute the first few powers of M: \(M^2 =\left[\begin{array}{ll}{1} & {1} \\\ {1} & {0}\end{array}\right]\left[\begin{array}{ll}{1} & {1} \\\ {1} & {0}\end{array}\right]=\left[\begin{array}{ll}{2} & {1} \\\ {1} & {1}\end{array}\right]\) \(M^3 = M \cdot M^2 = \left[\begin{array}{ll}{1} & {1} \\\ {1} & {0}\end{array}\right]\left[\begin{array}{ll}{2} & {1} \\\ {1} & {1}\end{array}\right]=\left[\begin{array}{ll}{3} & {2} \\\ {2} & {1}\end{array}\right]\) Observe a pattern with the computed powers of M: \(M^n =\left[\begin{array}{ll}{F_{n+1}} & {F_n} \\\ {F_n}& {F_{n-1}}\end{array}\right] \) where \(F_n\) is the nth Fibonacci number. For example, when \(n=2\), we have the Fibonacci numbers as \(F_{3}=2\), \(F_{2}=1\), \(F_{1}=1\), and \(F_{0}=0\).

Step 3: Compute f(4)

Using the pattern observed in Step 2, we have: \(f(4) = M^4 = \left[\begin{array}{ll}{F_{4+1}} & {F_4} \\\ {F_4} & {F_{3}}\end{array}\right]\) To find the Fibonacci numbers, use the definition: \(F_n = F_{n-1} + F_{n-2}\) for \(n \ge 2\), where \(F_0 = 0\) and \(F_1 = 1\). \(F_2 = 1\), \(F_3 = F_1 + F_2 = 2\), and \(F_4 = F_2 + F_3 = 3\). Therefore, \(f(4) = M^4 =\left[\begin{array}{ll}{F_{5}} & {F_4} \\\ {F_4} & {F_{3}}\end{array}\right]= \left[\begin{array}{ll}{5} & {3} \\\ {3} & {2}\end{array}\right]\)

Key Concepts

Fibonacci Sequence

The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding numbers. The sequence starts with 0 and 1, and it follows this pattern: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. This sequence is essential across various fields for its mathematical properties and applications. Fibonacci numbers appear in nature, for instance, the arrangement of leaves, the fruit sprouts of a pineapple, and the branching of trees.
In mathematics, especially in solving problems using matrix methods, the Fibonacci sequence can be embedded within matrix powers. For example, by taking powers of a specific 2x2 matrix, known as the Fibonacci matrix, we can generate Fibonacci numbers. This is why in the given exercise, the matrix power leads to values that align with the Fibonacci sequence.

Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices by each other. Unlike number multiplication, the product of two matrices is not always commutative, meaning matrix A times matrix B is different from matrix B times matrix A. The number of columns in the first matrix must match the number of rows in the second matrix for the multiplication to be defined.
Here's a brief guide on how matrix multiplication works. Given matrices \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) and \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix} \).The product \(A \cdot B \) will be:\[ \begin{bmatrix} (ae + bg) & (af + bh) \ (ce + dg) & (cf + dh) \end{bmatrix} \]Matrix multiplication comes in handy when exploring patterns in discrete mathematics; for instance, the Fibonacci sequence in the context of powers of matrices like in our exercise. It involves repeatedly multiplying the Fibonacci matrix by itself.

Discrete Mathematics

Discrete Mathematics is a branch of mathematics dealing with discrete rather than continuous quantities. It covers topics like graphs, logic, algorithms, and sequences—of which the Fibonacci sequence is a part. In computer science and mathematics, discrete mathematics is crucial due to its application in algorithm development, particularly for solving problems that involve distinct and separate values.
In the context of the exercise, discrete mathematics allows us to model and compute the powers of matrices easily and recognize patterns that are not continuous, such as the Fibonacci sequence. Using matrices to solve problems that involve the Fibonacci sequence is a prime example of how discrete mathematics operates.
The power of matrices in discrete mathematics helps us simplify complex sequence problems, like identifying Fibonacci numbers in higher terms without calculating all the previous ones separately, thus saving computation time and resources.