/*! 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 7 Find the dimensions of the given... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 7

Find the dimensions of the given matrix and identify the given entry. $$ B=\left[\begin{array}{rr} 1 & 3 \\ 5 & -6 \end{array}\right] ; b_{12} $$

Short Answer

Expert verified
The matrix B has dimensions 2x2, and the entry \(b_{12}\) is located in the first row and second column, which is the value 3.

Step by step solution

01

Identify the number of rows and columns in matrix B

Count the number of rows and columns in the given matrix B. The rows are the horizontal lines and the columns are the vertical lines.
02

Determine the dimensions of matrix B

After counting the number of rows and columns in matrix B, we find that there are 2 rows and 2 columns. Thus, the dimensions of matrix B are 2x2.
03

Identify the entry \(b_{12}\) in matrix B

In the given matrix entry \(b_{12}\), the first index (1) represents the row number, and the second index (2) represents the column number. Therefore, \(b_{12}\) is the element located in the first row and second column of matrix B.

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 Identification
When dealing with matrices, it's essential to know how to identify them properly. Identification begins with observing the overall layout of the entries:
  • A matrix is essentially a rectangular array or grid of numbers.
  • It is organized into rows and columns, similar to a spreadsheet.
  • The number of rows is counted horizontally, whereas columns are counted vertically.
  • Identifying a matrix also includes noting its dimensions, described by the number of rows and columns it contains.
In identifying our matrix B, it consists of a 2x2 configuration, meaning there are 2 rows and 2 columns. Understanding these basics is crucial for further operations such as matrix addition, subtraction, and multiplication.
Matrix Notation
Matrix notation is the language we use to describe the specific elements and arrangement within a matrix. Here鈥檚 a simple breakdown:
  • Each matrix is often denoted by a capital letter, such as matrix B in our example.
  • Entries within the matrix can be referenced using subscripts. For instance, the notation \(b_{ij}\) refers to an element in the i-th row and the j-th column.
  • This helps in pinpointing exactly where each number is situated, which is crucial for performing any matrix operations.
In matrix B:
  • expressions like \(b_{12}\) indicate you are referring to the element in the first row and second column.
Using consistent notation allows easy communication of mathematical ideas among learners and educators alike.
Matrix Entries
Matrix entries are the individual numbers that make up the matrix. These are the key to understanding the matrix鈥檚 application.
  • They can be integers, real numbers, or even complex numbers. For our purposes, we'll stick with the realm of real numbers.
  • Each entry can affect calculations such as determinants or inversions when notated and utilized correctly.
  • The value and position of these entries are instrumental in solving systems of equations and transforming geometric representations.
Let鈥檚 examine the entry \(b_{12}\) in matrix B:
  • This entry, which is 3, resides at the intersection of the first row and second column.
  • Grasping the role of this entry is essential for understanding the structure and operations performed on the matrix.
By focusing on entries and their respective locations, students can effectively utilize matrices in mathematical 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

Can the payoff in a saddle point ever be larger than all other payoffs in a game? Explain.

If a point \((x, y)\) in the plane is rotated counterclockwise through an angle of \(60^{\circ}\), its new coordinates are given by $$ \left[\begin{array}{l} x^{\prime} \\ y^{\prime} \end{array}\right]=S\left[\begin{array}{l} x \\ y \end{array}\right] $$ a. If the point \((2,3)\) is rotated counterclockwise through an angle of \(60^{\circ}\), what are its (approximate) new coordinates? b. Referring to Exercise 61, multiplication by what matrix would result in a counterclockwise rotation of \(105^{\circ} ?\) (Express the matrices in terms of \(S\) and the matrix \(R\) from Exercise 61.) [Hint: Think of a rotation through \(105^{\circ}\) as a rotation through \(60^{\circ}\) followed by one through \(45^{\circ}\).] c. Multiplication by what matrix would result in a clockwise rotation of \(60^{\circ}\) ?

Revenue Recall the Left Coast Bookstore chain from the preceding section. In January, it sold 700 hardcover books, 1,300 softcover books, and 2,000 plastic books in San Francisco; it sold 400 hardcover, 300 softcover, and 500 plastic books in Los Angeles. Now, hardcover books sell for \(\$ 30\) each, softcover books sell for \(\$ 10\) each, and plastic books sell for \(\$ 15\) each. Write a column matrix with the price data and show how matrix multiplication (using the sales and price data matrices) may be used to compute the total revenue at the two stores.

Calculate (a) \(P^{2}=P \cdot P\) (b) \(P^{4}=P^{2} \cdot P^{2}\) and \(\left(\right.\) c) \(P^{8} .\) Round all entries to four decimal places.) (d) Without computing it explicitly, find \(P^{1000}\). $$ P=\left[\begin{array}{lll} 0.3 & 0.3 & 0.4 \\ 0.3 & 0.3 & 0.4 \\ 0.3 & 0.3 & 0.4 \end{array}\right] $$

(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?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Statistics

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.