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Chapter 8: Problem 61

Matrix G gives the U.S. gross domestic product for the years \(1999-2001\) GDP(billions of \(\mathfrak{S})$$\begin{array}{l}1999 \\ 2000 \\\ 2001\end{array}\left[\begin{array}{r}9274.3 \\ 9824.6 \\\ 10,082.2\end{array}\right]=G\) The finance, retail, and agricultural sectors contributed \(20 \%, 9 \%,\) and \(1.4 \%,\) respectively, to the gross domestic product in those years. These percentages have been converted to decimals and are given in matrix \(P .\) (Source: U.S. Bureau of Economic Analysis) Finance Retail Agriculture $$\left[\begin{array}{lll}0.2 & 0.09 & 0.014\end{array}\right]=P$$ (a) Compute the product \(G P\). (b) What does GP represent? (c) Is the product \(P G\) defined? If so, does it represent anything meaningful? Explain.

Short Answer

Expert verified
The product \(GP\) represents the amount contributed by each sector (Finance, Retail, Agriculture) to the GDP for each year, and cannot be computed for the product \(PG\) since their respective rows and columns do not satisfy the conditions for matrix multiplication.

Step by step solution

01

Compute the product GP

Matrix multiplication rules dictate that to multiply two matrices, the number of columns in the first matrix should be equal to the number of rows in the second matrix. Here, both matrix G and P satisfy this requirement, and so we can compute the product as follows:\n\n\( GP = \begin{array}{r}9274.3 \ 9824.6 \ 10,082.2\end{array} \times \begin{array}{r}0.2 & 0.09 & 0.014\end{array} \)\n\nAccording to the rule of matrix multiplication, each element in the resulting matrix is the product of respective row from the first matrix and column of the second matrix. Compute these products and sum them for each year.
02

Interpret the product

The resulting product represents the amount (in billions) contributed by each sector (finance, retail, and agriculture) to the GDP each year. Each element in the product matrix corresponds to the contribution of a particular sector for a given year.
03

Check whether the product PG is defined

For the Matrix P with dimensions 1*3, and Matrix G with dimensions 3*1 to be multiplied as \(PG\), the number of columns in P should be equal to the number of rows in G. This condition is not satisfied here, so the product PG is not defined.
04

Interpret the undefined product

Since the product is not defined, it does not represent any meaningful economic quantities in this context.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Gross Domestic Product
Gross Domestic Product (GDP) is a key indicator that measures the total value of all goods and services produced in a country over a specific period, usually annually or quarterly. It's an important metric because it provides a comprehensive view of a country's economic activity and health.
When we talk about GDP, we refer to the combined output of a nation's industries and sectors. In this exercise, we see GDP represented through a matrix, showcasing data for the years 1999, 2000, and 2001.
GDP is crucial for:
  • Comparing economic productivity over different periods.
  • Analyzing the impact of financial policies and economic reforms.
  • Understanding the market value of an economy.
Matrix representation helps organize this data in a straightforward and understandable manner, making it easier to perform calculations like sector contributions.
Exploring Matrix Operations in Economics
Matrix operations are fundamental in mathematics, especially in handling complex calculations involving large datasets. In economics, matrices are used for various applications including the analysis of economic data like GDP contributions from different sectors.
One of the key operations here is matrix multiplication. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. In our scenario, matrix G (representing GDP values) is multiplied by matrix P (sector contribution percentages).
The multiplication process involves:
  • Taking a row from matrix G and a column from matrix P.
  • Multiplying corresponding entries together.
  • Summing these products to get a single entry in the resulting matrix.
This operation results in a new matrix that details the contributions of finance, retail, and agriculture sectors to the GDP over the given years. Understanding these steps and logic allows you to see how interventions affect economic sectors and overall economic health.
Sector Contributions to GDP through Matrix Multiplication
The sector contributions matrix (P) offers insight into the percentage each sector contributes to the overall GDP. By converting these percentages into decimals and organizing them into a matrix, we can systematically distribute GDP data across different sectors.
In the exercise, when you multiply the GDP matrix (G) by the sector contribution matrix (P), you achieve a breakdown of how much each sector contributes in monetary terms over the years 1999 to 2001.
Here's what the matrix GP tells us:
  • Each element in the resultant matrix from multiplying G and P represents the financial contribution of a specific sector to the GDP in a particular year.
  • Understanding these nuances aids policymakers and analysts in identifying which sectors are most vital to economic growth and stability.
Essentially, this process of using matrices to calculate sector contributions provides a clearer picture of economic segmentation and efficiency, paving the way for more informed economic planning and resource allocation.

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Most popular questions from this chapter

In this set of exercises, you will use the method of solving linear systems using matrices to study real-world problems. Electrical Engineering An electrical circuit consists of three resistors connected in series. The formula for the total resistance \(R\) is given by \(R=R_{1}+R_{2}+R_{3},\) where \(R_{1}, R_{2},\) and \(R_{3}\) are the resistances of the individual resistors. In a circuit with two resistors \(A\) and \(B\) connected in series, the total resistance is 60 ohms. The total resistance when \(B\) and \(C\) are connected in series is 100 ohms. The sum of the resistances of \(B\) and \(C\) is 2.5 times the resistance of \(A\). Find the resistances of \(A, B\), and \(C\).

In this set of exercises, you will use the method of solving linear systems using matrices to study real-world problems. The athletic director of a local high school is ordering equipment for spring sports. He needs to order twice as many baseballs as softballs. The total number of balls he must order is \(300 .\) How many of each type should he order?

Keith and two of his friends, Sam and Cody, take advantage of a sidewalk sale at a shopping mall. Their purchases are summarized in the following table. $$\begin{array}{lc|c|c|} \hline& {3}{|}\text { Quantity } \\\\\hline\text { Name } & \text { Shirt } & \text { Sweater } & \text { Jacket } \\\\\hline \text { Keith } & 3 & 2 & 1 \\\\\text { Sam } & 1 & 2 & 2 \\\\\text { Cody } & 2 & 1 & 2\\\\\hline\end{array}$$ The sale prices are \(\$ 14.95\) per shirt, \(\$ 18.95\) per sweater, and \(\$ 24.95\) per jacket. In their state, there is no sales tax on purchases of clothing. Use matrix multiplication to determine the total expenditure of each of the three shoppers.

Let \(I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] .\) Show that \(I A=A I,\) where \(A\) is any \(2 \times 2\) matrix.

This set of exercises will draw on the ideas presented in this section and your general math background. Compute \(A(B C)\) and \((A B) C,\) where \(A=\left[\begin{array}{rr}3 & -1 \\ 0 & 2\end{array}\right], \quad B=\left[\begin{array}{ll}1 & 4 \\ 0 & 1\end{array}\right], \quad\) and \(\quad C=\left[\begin{array}{rr}-1 & 0 \\ 3 & 1\end{array}\right]\) What do you observe?
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