Problem 69 - College Algebra and Calculus: An Applied Approach

Question

The matrix $$ P=\left[\begin{array}{lll} 0.6 & 0.1 & 0.1 \\ 0.2 & 0.7 & 0.1 \\ 0.2 & 0.2 & 0.8 \end{array}\right] $$ is called a stochastic matrix. Each entry \(p_{i j}(i \neq j)\) represents the proportion of the voting population that changes from Party \(i\) to Party \(j\), and \(p_{i i}\) represents the proportion that remains loyal to the party from one election to the next. Use a graphing utility to find \(P^{2}\). (This matrix gives the transition probabilities from the first election to the third.)

Step-by-Step Solution

Answer

The resulting matrix \( P^2 \) will be a 3x3 matrix, with each element calculated as described in step 2.

Step 1: Write down the given matrix

The given matrix is \( P = \begin{bmatrix} 0.6 & 0.1 & 0.1 \\ 0.2 & 0.7 & 0.1 \\ 0.2 & 0.2 & 0.8 \end{bmatrix} \).

Step 2: Perform the matrix multiplication

To get \( P^2 \), multiply the matrix P with itself. Calculating element by element,\( (P^2)_{11} = P_{11}*P_{11} + P_{12}*P_{21} + P_{13}*P_{31} = 0.6*0.6 + 0.1*0.2 + 0.1*0.2 = 0.38 \),\( (P^2)_{12} = P_{11}*P_{12} + P_{12}*P_{22} + P_{13}*P_{32} = 0.6*0.1 + 0.1*0.7 + 0.1*0.2 = 0.17 \),and so on for each element in the matrix P^2.

Step 3: Write down the calculated matrix

After calculating each of the elements, write down the resulting matrix \( P^2 \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra. It allows us to compute the result of sequential transformations, to combine different linear mappings into one, or, in terms of probability, determine composite transition probabilities. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second. If we have a matrix A of size m by n and a matrix B of size n by p, their product, AB, will be a new matrix of size m by p.

For each element of the resulting matrix, we calculate the sum of the products of corresponding entries from the row of the first matrix and the column of the second matrix. This can be expressed mathematically by the equation: \( (AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} \).

When multiplying a stochastic matrix like P with itself to find \( P^2 \), each element in \( P^2 \) can be thought of as the probability of transition from one state to another over two steps. This kind of multiplication is especially common in the study of Markov chains where such processes describe state changes over time.

Transition Probabilities

Transition probabilities represent the likelihood of moving from one state to another in a certain system or process. In the context of voting populations, each element \( p_{ij} \) of a stochastic matrix P represents the proportion of the population that transitions from one party to another between elections. Diagonal elements \( p_{ii} \), where the row and column indices are the same, represent the proportion of the population that remains loyal to their current party.

When we calculate \( P^2 \), we're looking at the probabilities after two transitions, which can predict longer-term trends based on initial states. For instance, the element \( (P^2)_{ij} \) provides the probability that voters loyal to Party i will be with Party j after two elections, accounting for all possible intermediate transitions. Understanding these probabilities is crucial for forecasting and modeling behavior in complex systems.

Using a Graphing Utility

A graphing utility is an integral tool for visual learners and efficient computation. It can be particularly useful when dealing with matrices and their operations, such as raising a stochastic matrix to the power of 2 to find \( P^2 \). Most modern graphing utilities can perform complex calculations, like matrix multiplication, within seconds.

This not only saves time but also reduces the likelihood of manual calculation errors. After the calculation is completed, these utilities often offer the capability to graphically represent data, which can be extremely helpful for interpreting matrix operations in a spatial or geometrical context, such as visualizing the flow of transitions between parties in our voting example.