/*! 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 1 Compute the products. Some of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 1

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. \(\left[\begin{array}{lll}1 & 3 & -1\end{array}\right]\left[\begin{array}{r}9 \\\ 1 \\ -1\end{array}\right]\)

Short Answer

Expert verified
The product of the given matrices is \(13\).

Step by step solution

01

Understanding matrix multiplication rules

Matrix multiplication of two matrices is only possible if the number of columns in the first matrix (in our case, 3) is equal to the number of rows in the second matrix (also 3). Since this condition is met, we can proceed with matrix multiplication.
02

Perform the multiplication by hand

To perform the multiplication, we need to multiply each element of the row matrix with the corresponding element from the column matrix and then add the products. \( \left[\begin{array}{lll}1 & 3 & -1\end{array}\right]\left[\begin{array}{r}9\\\ 1 \\\ -1\end{array}\right]=(1\cdot9)+(3\cdot1)+(-1\cdot(-1)) \) Now, calculate the expression: \( =(1\cdot9)+(3\cdot1)+(-1\cdot(-1))=9+3+1 \) Add the numbers: \( =13 \)
03

Perform the multiplication using technology

It is also possible to perform this matrix multiplication using technology, such as a calculator with matrix capabilities, a spreadsheet program, or a programming language like Python with the NumPy library. Regardless of the tool, the result should be the same: 13.

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.

Matrix Operations
Matrix operations are fundamental in the study of computational mathematics and linear algebra. They include addition, subtraction, and multiplication, as well as more complex operations like inversion and transposition. Matrix multiplication, in particular, is a core operation where two matrices are multiplied to produce a third matrix. Understanding when and how you can multiply matrices is important:
  • The number of columns in the first matrix must equal the number of rows in the second matrix.
  • The resulting matrix will have the dimensions of the number of rows of the first matrix by the number of columns of the second matrix.
In our exercise, we multiply a row vector with a column vector, both with matching dimensions, leading to a single numerical result, a special case of matrix multiplication. In practical applications, multiplying matrices is used in graphics transformations, scientific computations, and solving systems of linear equations.
Linear Algebra
Linear algebra is a field of mathematics that deals with vector spaces and linear mappings between these spaces. It is broadly used in engineering, physics, natural sciences, and computer science due to its versatility and extensive application range. The exercise at hand utilizes a basic concept from linear algebra - the multiplication of matrices.
Matrix multiplication is a linear transformation that can represent more complex transformations when combined. In our exercise:
  • The single row, which can be viewed as a horizontal vector, represents a set of linear equations.
  • The single column, a vertical vector, acts as directional multipliers or coefficients.
This multiplication gives a simplification and transformation result, seen directly in the scalar value output, demonstrating the power and simplicity of linear transformations in reducing complex data structures to more manageable forms. Understanding these basic operations builds the foundation for tackling more complex problems in linear algebra.
Computational Mathematics
Computational mathematics involves using algorithms and numerical methods to solve mathematical problems that are computationally intensive or practically unsolvable by hand. In matrix operations, including those in our example:
  • Computational tools such as Python's NumPy library or dedicated calculators can handle large computations rapidly and accurately.
  • Technology aids in visualizing data transformations, understanding behaviors across large datasets, and facilitating simulations.
While our exercise can be easily calculated by hand, for larger matrices or repeated operations, technology can significantly enhance learning efficiency and computational power. Interacting with linguistic tools and programming enhances comprehension of mathematical concepts and equips users with invaluable skills for the computational tasks in fields like data analysis, quantum computing, and artificial intelligence.

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

Decide whether the game is strictly determined. If it is, give the players'optimal pure strategies and the value of the game. $$ \begin{gathered} \text { B } \\ p & q \\ \text { A } \left.\begin{array}{rr} a \\ 1 & 1 \\ 2 & -4 \end{array}\right] \end{gathered} $$

Textbook Writing You are writing a college-level textbook on finite mathematics, and are trying to come up with the best combination of word problems. Over the years, you have accumulated a collection of amusing problems, serious applications, long complicated problems, and "generic" problems. \({ }^{25}\) Before your book is published, it must be scrutinized by several reviewers who, it seems, are never satisfied with the mix you use. You estimate that there are three kinds of reviewers: the "no-nonsense" types who prefer applications and generic problems, the "dead serious" types, who feel that a collegelevel text should be contain little or no humor and lots of long complicated problems, and the "laid-back" types, who believe that learning best takes place in a light-hearted atmosphere bordering on anarchy. You have drawn up the following chart, where the payoffs represent the reactions of reviewers on a scale of \(-10\) (ballistic) to \(+10\) (ecstatic): Reviewers ou \begin{tabular}{|l|c|c|c|} \hline & No-Nonsense & Dead Serious & Laid-Back \\ \hline Amusing & \(-5\) & \(-10\) & 10 \\ \hline Serious & 5 & 3 & 0 \\ \hline Long & \(-5\) & 5 & 3 \\ \hline Generic & 5 & 3 & \(-10\) \\ \hline \end{tabular} a. Your first draft of the book contained no generic problems, and equal numbers of the other categories. If half the reviewers of your book were "dead serious" and the rest were equally divided between the "no-nonsense" and "laid-back" types, what score would you expect? b. In your second draft of the book, you tried to balance the content by including some generic problems and eliminating several amusing ones, and wound up with a mix of which one eighth were amusing, one quarter were serious, three eighths were long, and a quarter were generic. What kind of reviewer would be least impressed by this mix? c. What kind of reviewer would be most impressed by the mix in your second draft?

What would it mean if the technology matrix \(A\) were the zero. matrix?

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. \({ }^{14}\) 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.) $$ \left.\begin{array}{ccccc} & \text { To } & \text { To } & \text { To } & \text { To } \\ & \text { NE } & \text { MW } & \text { S } & \text { W } \\ \text { From NE } & 0.9892 & 0.0017 & 0.0073 & 0.0018 \\ \text { P }= & \text { From MW } & 0.0010 & 0.9920 & 0.0048 & 0.0022 \\ \text { From S } & 0.0018 & 0.0024 & 0.9934 & 0.0024 \\ \text { From W } & 0.0008 & 0.0033 & 0.0045 & 0.9914 \end{array}\right] $$ Set up the 2006 population figures as a row vector. Then use matrix multiplication to compute the population in each region in 2007 . (Round all answers to the nearest \(0.1\) million.)

A diagonal matrix \(D\) has the following form. $$ D=\left[\begin{array}{ccccc} d_{1} & 0 & 0 & \ldots & 0 \\ 0 & d_{2} & 0 & \ldots & 0 \\ 0 & 0 & d_{3} & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & d_{n} \end{array}\right] $$ When is \(D\) singular? Why?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.