Problem 27 - Precalculus

Question

Consider the following scenario. In the small village of Pedimaxus in the country of Sasquatchia, all 150 residents get one of the two local newspapers. Market research has shown that in any given week, \(90 \%\) of those who subscribe to the Pedimaxus Tribune want to keep getting it, but \(10 \%\) want to switch to the Sasquatchia Picayune. Of those who receive the Picayune, \(80 \%\) want to continue with it and \(20 \%\) want switch to the Tribune. We can express this situation using matrices. Specifically, let \(X\) be the 'state matrix' given by $$ X=\left[\begin{array}{l} T \\ P \end{array}\right] $$ where \(T\) is the number of people who get the Tribune and \(P\) is the number of people who get the Picayune in a given week. Let \(Q\) be the 'transition matrix' given by $$ Q=\left[\begin{array}{ll} 0.90 & 0.20 \\ 0.10 & 0.80 \end{array}\right] $$ such that \(Q X\) will be the state matrix for the next week. If you didn't see the pattern, we'll help you out. Let $$ X_{s}=\left[\begin{array}{r} 100 \\ 50 \end{array}\right]. $$ Show that \(Q X_{s}=X_{s}\) This is called the steady state because the number of people who get each paper didn't change for the next week. Show that \(Q^{n} X \rightarrow X_{s}\) as \(n \rightarrow \infty\).

Step-by-Step Solution

Answer

The steady state, \( X_s = \begin{bmatrix} 100 \\ 50 \end{bmatrix} \), remains unchanged under transition matrix \( Q \).

Step 1: Understand Initial Variables and Matrices

The village has 150 residents, and the state matrix \( X \) represents the number of people subscribing to each newspaper. \( T \) is the number of Tribune subscribers, and \( P \) is the number of Picayune subscribers. \( Q \) is the transition matrix that indicates the probability of subscribers switching papers.

Step 2: Compute Transition for One Week

To find next week's distribution of subscribers, multiply the transition matrix \( Q \) by the initial state matrix \( X_s \): \[QX_s = \begin{bmatrix} 0.90 & 0.20 \ 0.10 & 0.80 \end{bmatrix} \begin{bmatrix} 100 \ 50 \end{bmatrix}\]. This results in the matrix \( \begin{bmatrix} 100 \ 50 \end{bmatrix} \).

Step 3: Verify No Change in One Week

Calculate the matrix multiplication: \[\begin{bmatrix} 0.90 & 0.20 \ 0.10 & 0.80 \end{bmatrix} \begin{bmatrix} 100 \ 50 \end{bmatrix} = \begin{bmatrix} (0.90 \times 100) + (0.20 \times 50) \ (0.10 \times 100) + (0.80 \times 50) \end{bmatrix} = \begin{bmatrix} 90 + 10 \ 10 + 40 \end{bmatrix} = \begin{bmatrix} 100 \ 50 \end{bmatrix}\].

Step 4: Show Steady State Over Time

Repeated application of \( Q \) to \( X \) (i.e., multiplying successively) will keep resulting in \( \begin{bmatrix} 100 \ 50 \end{bmatrix} \). This means as \( n \to \infty \), \( Q^n X \to X_s \), demonstrating a steady state where state doesn't change.

Key Concepts

Understanding Matrices

Matrices are powerful mathematical tools that help organize and manipulate data in a structured form. In our scenario, we use matrices to represent and understand the behavior of newspaper subscribers in a village. A matrix is essentially a rectangular array of numbers arranged in rows and columns. Each matrix element represents a specific piece of data. Here, the state matrix and the transition matrix are key components for analyzing the subscription patterns.
For example, the state matrix in our problem is a vertical matrix with two entries, representing the number of Tribune and Picayune subscribers. The transition matrix is a square matrix that shows the probabilities of subscribers switching from one newspaper to the other. These matrices work together to predict how the number of subscribers changes over time.

Introducing the Transition Matrix

A transition matrix is a special type of square matrix used to describe the probabilities of shifting between different states. It's essentially a mathematical way of showing possible changes over time. In the Pedimaxus village scenario, the transition matrix \( Q \) demonstrates the chances of residents switching from one newspaper to another.
This matrix is essential because it allows us to predict the future distribution of subscribers when multiplied by the current state matrix. The entries of \( Q \) tell us the likelihood of an individual switching or staying with their current newspaper.

• Entry \( Q_{T,T} = 0.90 \): Probability of continuing with Tribune
• Entry \( Q_{T,P} = 0.20 \): Probability of switching to Tribune from Picayune
• Entry \( Q_{P,T} = 0.10 \): Probability of switching to Picayune from Tribune
• Entry \( Q_{P,P} = 0.80 \): Probability of continuing with Picayune

By using the transition matrix, we can calculate the number of subscribers for each newspaper, predicting not only next week's subscribers but also understanding trends over a longer period.

Discovering the State Matrix

The state matrix is an important component when dealing with systems that evolve over time. It represents the current distribution of different states in the system. In our exercise, it's a way to encapsulate the number of subscribers to each newspaper at any given time.
For instance, if the state matrix initially is \( X_s = \begin{bmatrix} 100 \ 50 \end{bmatrix} \), it means 100 people subscribe to the Tribune and 50 to the Picayune. When multiplied by the transition matrix, it tells us how these numbers are expected to change over a week.
By continuously updating the state matrix through this multiplication, we can track how the subscriber numbers transition week by week. This state matrix, when reaching a point where further multiplication does not change its values, showcases a steady state, a point of equilibrium for the system.

Exploring Steady State Over Time

In mathematical modeling, especially with matrices, the steady state is a crucial concept that describes a condition where the system no longer changes despite any transition activities. In simpler terms, after applying the transition matrix multiple times, the state matrix reaches a point where it stays constant. This point is the steady state.
Let's apply this to our village newspaper scenario. When we multiply the transition matrix \( Q \) by the initial state matrix \( X_s \), we might continue seeing the same state matrix \( \begin{bmatrix} 100 \ 50 \end{bmatrix} \). This confirms that this system has reached its steady state.
The steady state reflects a balance where the inflow and outflow of subscribers between the newspapers equalizes. This observation gives insights into long-term trends and how resilient certain subscriber patterns are to change. In terms of application, it tells us that regardless of initial conditions, the system will invariably stabilize to this steady distribution over time.