/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Compute the products. Some of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 24

Compute the products. Some of these may be undefined.Exercises marked I should be done using technology. The others should be done two ways: by hand and by using technology where possible. \(\left[\begin{array}{rrr}1 & 2 & 0 \\ 4 & -1 & 1 \\ 1 & 0 & 1\end{array}\right]\left[\begin{array}{rrr}1 & 2 & -4 \\ 4 & 1 & 0 \\ 0 & -2 & 1\end{array}\right]\)

Short Answer

Expert verified
The product of the given matrices is \(AB = \left[\begin{array}{rrr} 9 & 4 & -4 \\\ 0 & 7 & -15 \\\ 1 & 0 & -3\end{array}\right]\).

Step by step solution

01

Set up the problem

To multiply the matrices, we'll use the dot product method. Let matrix A be \(\left[\begin{array}{rrr}1 & 2 & 0 \\\ 4 & -1 & 1 \\\ 1 & 0 & 1\end{array}\right]\) and matrix B be \(\left[\begin{array}{rrr}1 & 2 & -4 \\\ 4 & 1 & 0 \\\ 0 & -2 & 1\end{array}\right]\). Our goal is to find the product AB, which will also be a 3x3 matrix.
02

Multiply the matrices by hand

To find the dot product of two matrices, we will multiply the elements in the rows of matrix A by the elements in the columns of matrix B and sum the products. For example, to find the first element (top-left) of the resulting matrix, we perform the following operation: \((1\times1)+(2\times4)+(0\times0)=1+8=9\) Computing all the elements, we find: \(AB = \left[\begin{array}{rrr} (1\times1)+(2\times4)+(0\times0) & (1\times2)+(2\times1)+(0\times(-2)) & (1\times(-4))+(2\times0)+(0\times1) \\\ (4\times1)+(-1\times4)+(1\times0) & (4\times2)+(-1\times1)+(1\times(-2)) & (4\times(-4))+(-1\times0)+(1\times1) \\\ (1\times1)+(0\times4)+(1\times0) & (1\times2)+(0\times1)+(1\times(-2)) & (1\times(-4))+(0\times0)+(1\times1) \end{array}\right]\) \(AB = \left[\begin{array}{rrr} 9 & 4 & -4 \\\ 0 & 7 & -15 \\\ 1 & 0 & -3\end{array}\right]\)
03

Multiply the matrices using technology

There are many ways to multiply matrices using technology. For example, you can use a calculator with matrix functionality, an online matrix calculator, or a computer algebra system (such as Mathematica, MATLAB, or Python with NumPy). For this solution, we will use the numpy library in Python: ```python import numpy as np A = np.array([[1, 2, 0], [4, -1, 1], [1, 0, 1]]) B = np.array([[1, 2, -4], [4, 1, 0], [0, -2, 1]]) result = np.matmul(A,B) print(result) ``` This will output: ``` [[ 9 4 -4] [ 0 7 -15] [ 1 0 -3]] ``` Both methods give us the same result, which is the product of the given matrices: \(AB = \left[\begin{array}{rrr} 9 & 4 & -4 \\\ 0 & 7 & -15 \\\ 1 & 0 & -3\end{array}\right]\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Dot Product Method
The dot product method is fundamental in matrix multiplication and plays a critical role in various applications such as computer graphics, economics, and applied mathematics. When we multiply two matrices, we calculate the dot product of rows from the first matrix with columns from the second matrix to form the resulting matrix.

To compute the dot product, you multiply corresponding entries from the row and column and add up those products. For instance, if we want to find the result of the first entry of the product matrix, we take the first row of the first matrix and the first column of the second matrix, then calculate as \( (1 \times 1) + (2 \times 4) + (0 \times 0) = 9 \).

You would repeat this process for each entry in the result matrix, systematically working through all the rows of the first matrix and the columns of the second matrix until the entire result matrix is filled out.
Matrix Product by Hand
Multiplying matrices by hand provides valuable practice in understanding the mechanics behind the process. It's important to ensure that the number of columns in the first matrix matches the number of rows in the second matrix; otherwise, the product is undefined.

Using the given matrices from the example, you would calculate each entry of the resulting matrix one at a time using the dot product method described above. This manual method involves a lot of systematic arithmetic work, and while it can be quite time-consuming, especially for larger matrices, it helps to reinforce the fundamental concepts of matrix operations.
Matrix Product Using Technology
Nowadays, technology can significantly speed up the process of matrix multiplication. There are various technological tools available, from calculators with matrix functions to sophisticated software like MATLAB or Python libraries.

These tools can perform the same multiplications in a fraction of the time it takes to compute by hand, reducing the possibility of human error in the calculations. By using these technologies, students and professionals can focus on the application and interpretation of the results rather than the computation itself. It is still recommended to understand how the multiplication works manually before relying solely on technology.
Numpy Library in Python
The NumPy library in Python is a powerful tool for numerical computing and is particularly well-suited for handling arrays and matrices. NumPy's functions are highly optimized and can execute complex matrix operations rapidly.

For instance, the np.matmul() function or the @ operator can be used to multiply two matrices. When using NumPy to multiply matrices from the example, the code is relatively simple and gives us an immediate result that is accurate and efficiently calculated. Python's simplicity and the power of the NumPy library make it a popular choice for performing matrix operations in fields that require quick and reliable computations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Australia Economy \(^{34}\) Two sectors of the Australian economy are (1) community services and (2) recreation services. The 1978-79 input-output table involving these two sectors results in the following value for \((I-A)^{-1}\) : $$ (I-A)^{-1}=\left[\begin{array}{ll} 1.0066 & 0.00576 \\ 0.00496 & 1.04206 \end{array}\right] $$ Complete the following sentences. a. \(0.00496\) additional dollars worth of \(\quad\) must be produced to meet a \(\$ 1\) increase in the demand for b. additional dollars worth of community services must be produced to meet a one-dollar increase in the demand for community services.

I Population Movement In 2006, the population of the United States, broken down by regions, was \(55.1\) million in the Northeast, \(66.2\) million in the Midwest, \(110.0\) million in the South, and \(70.0\) million in the West. \({ }^{19}\) The matrix \(P\) below shows the population movement during the period 2006 \(2007 .\) (Thus, \(98.92 \%\) of the population in the Northeast stayed there, while \(0.17 \%\) of the population in the Northeast moved to the Midwest, and so on.) \(\begin{array}{ccll}\text { To } & \text { To } & \text { To } & \text { To } \\\ \text { NE } & \text { MW } & \text { S } & \text { W }\end{array}\) \(P=\begin{aligned}&\text { From NE } \\\&\text { From MW } \\\&\text { From S } \\\&\text { From W }\end{aligned}\left[\begin{array}{llll}0.9892 & 0.0017 & 0.0073 & 0.0018 \\ 0.0010 & 0.9920 & 0.0048 & 0.0022 \\ 0.0018 & 0.0024 & 0.9934 & 0.0024 \\ 0.0008 & 0.0033 & 0.0045 & 0.9914\end{array}\right]\) Set up the 2006 population figures as a row vector. Assuming that these percentages also describe the population movements from 2005 to 2006 , show how matrix inversion and multiplication allow you to compute the population in each region in 2005 . (Round all answers to the nearest \(0.1\) million.)

Translate the given systems of equations into matrix form. \(\begin{aligned} 2 x+y &=7 \\\\-x &=9 \end{aligned}\)

(Staff Cutbacks Frank Tempest manages a large snowplow service in Manhattan, Kansas, and is alarmed by the recent weather trends; there have been no significant snowfalls since 1993\. He is therefore contemplating laying off some of his workers, but is unsure about whether to lay off 5,10 , or 15 of his 50 workers. Being very methodical, he estimates his annual net profits based on four possible annual snowfall figures: 0 inches, 20 inches, 40 inches and 60 inches. (He takes into account the fact that, if he is running a small operation in the face of a large annual snowfall, he will lose business to his competitors because he will be unable to discount on volume.) a. During the past 10 years, the region has had 0 inches twice, 20 inches twice, 40 inches three times, and 60 inches three times. Based on this information, how many workers should Tempest lay off, and how much would it cost him? b. There is a \(50 \%\) chance that Tempest will lay off 5 workers and a \(50 \%\) chance that he will lay off 15 workers. What is the worst thing Nature can do to him in terms of snowfall? How much would it cost him? c. The Gods of Chaos (who control the weather) know that Tempest is planning to use the strategy in part (a), and are determined to hurt Tempest as much as possible. Tempest, being somewhat paranoid, suspects it too. What should he do?

Evaluate the given expression. Take \(A=\left[\begin{array}{rrr}1 & -1 & 0 \\\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]\), and \(C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .\) $$ B-C $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.