/*! 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 27 Compute the products. Some of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 27

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. $$ \left[\begin{array}{llrr} 1.1 & 2.3 & 3.4 & -1.2 \\ 3.4 & 4.4 & 2.3 & 1.1 \\ 2.3 & 0 & -2.2 & 1.1 \\ 1.2 & 1.3 & 1.1 & 1.1 \end{array}\right]\left[\begin{array}{ccc} -2.1 & 0 & -3.3 \\ -3.4 & -4.8 & -4.2 \\ 3.4 & 5.6 & 1 \\ 1 & 2.2 & 9.8 \end{array}\right] $$

Short Answer

Expert verified
The resulting matrix C after performing the matrix multiplication is: $$ \left[\begin{array}{ccc} 0.23 & 5.36 & -21.65 \\ -5.03 & 8.96 & -0.68 \\ 4.33 & 18.82 & -29.95 \\ -1.53 & 3.04 & 3.98 \end{array}\right] $$

Step by step solution

01

Check if matrix multiplication is possible

To check if the matrix multiplication is possible, let's count the number of columns in the first matrix and the number of rows in the second matrix: - The first matrix has 4 columns - The second matrix has 4 rows Since the number of columns in the first matrix is equal to the number of rows in the second matrix (4), we can multiply these two matrices.
02

Set up the resulting matrix

Let's set up the resulting matrix C, with its dimensions being determined by the dimensions of the two input matrices: - The first matrix has 4 rows - The second matrix has 3 columns So, the resulting matrix C will have 4 rows and 3 columns. Now, let's find the elements of this resulting matrix C.
03

Computing the elements of the resulting matrix

To compute the element C_ij, where i is the row number and j is the column number, we multiply the elements of the i-th row of the first matrix (A) by the elements of the j-th column of the second matrix (B), and sum up these products: C_ij = Sum(A_ik * B_kj) for k from 1 to 4 Performing these calculations for all elements of the resulting matrix C, we get the following values: C_11 = (1.1)*(-2.1) + (2.3)*(-3.4) + (3.4)*(3.4) + (-1.2)*(1) = -2.31 - 7.82 + 11.56 - 1.2 = 0.23 C_12 = (1.1)*(0) + (2.3)*(-4.8) + (3.4)*(5.6) + (-1.2)*(2.2) = -11.04 + 19.04 - 2.64 = 5.36 C_13 = (1.1)*(-3.3) + (2.3)*(-4.2) + (3.4)*(1) + (-1.2)*(9.8) = -3.63 - 9.66 + 3.4 - 11.76 = -21.65 (Calculate the rest of the elements similarly) The resulting matrix C is: $$ \left[\begin{array}{ccc} 0.23 & 5.36 & -21.65 \\ -5.03 & 8.96 & -0.68 \\ 4.33 & 18.82 & -29.95 \\ -1.53 & 3.04 & 3.98 \end{array}\right] $$ To verify the result using technology, you can use any matrix multiplication calculator or a programming language like Python with the Numpy library to perform the matrix multiplication and check if the result matches the one obtained manually.

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.

Finite Mathematics
In Finite Mathematics, matrix multiplication plays a crucial role in various applications such as economics, computer science, engineering, and the social sciences. Matrices provide a compact way of representing and manipulating data, which includes systems of equations, transformations, and much more.

When multiplying matrices, it's essential to understand the conditions under which multiplication is possible. In finite mathematics, if the first matrix has dimensions of m 脳 n (where m is the number of rows and n the number of columns), and the second matrix has dimensions of n 脳 p, then the matrices are said to be conformable for multiplication, resulting in a new matrix of dimensions m 脳 p.

Here's a simplified step of the matrix multiplication process:
  • Each element of the resulting matrix is calculated by taking the dot product of its corresponding row from the first matrix and column from the second matrix.
  • The 'dot product' means multiplying corresponding entries and summing those products.
  • This is summed up by the formula C_ij = 鈭� A_ik * B_kj for each element C_ij of the resulting matrix.
The verification of the resulting matrix can be done manually or through computational tools.
Applied Calculus
In the field of Applied Calculus, matrix multiplication often comes into play when dealing with linear transformations and systems of differential equations. A matrix can represent a linear transformation, and multiplication can be used to combine transformations or to apply a transformation to a vector.

Considering our example with multiplication, we see a practical application of matrices in transforming data. This can be visualized as transforming shapes in computer graphics or perhaps adjusting variables in an economic model. Applied Calculus concerns itself not just with the mechanics but also with understanding what these calculations represent in the real world.

While you may not need differential or integral calculus to multiply matrices, the concept of linearity is fundamental in calculus, and understanding matrices helps bolster that concept. Additionally, when experimenting with applied problems, particularly in physics or engineering, matrices often help to simplify complex systems into solvable equations using techniques you learn from calculus.
Computational Mathematics
Matrix multiplication is a cornerstone in the field of Computational Mathematics, where algorithms and numerical methods are designed to solve mathematic problems efficiently. The multiplication of large matrices is computationally intensive, and thus, numerous algorithms exist to optimize this process.

In our exercise, when multiplying the matrices by hand, we must consider the arithmetic intensity and the scope for errors in manual calculations. Computational tools are invaluable here鈥攅specially as matrices get larger or become part of larger computational systems.

Software libraries like Numpy in Python can handle matrix operations with ease, allowing us to check our work quickly or handle computations we cannot feasibly do by hand. The ability to algorithmically manipulate matrices with computers is a testament to the growth of computational mathematics and its applications across various scientific domains.

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

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?

Decide whether the game is strictly determined. If it is, give the players'optimal pure strategies and the value of the game. $$ \begin{array}{r} \mathbf{B} \\ p & q & r \\ a & {\left[\begin{array}{rrr} 2 & 0 & -2 \\ -1 & 3 & 0 \end{array}\right]} \end{array} $$

Campaign Strategies \(^{28}\) Florida and Ohio are "swing states" that have a large bounty of electoral votes and are therefore highly valued by presidential campaign strategists. Suppose it is now the weekend before Election Day 2008 , and each candidate (McCain and Obama) can visit only one more state. Further, to win the election, McCain needs to win both of these states. Currently McCain has a \(40 \%\) chance of winning Ohio and a \(60 \%\) chance of winning Florida. Therefore, he has a \(0.40 \times 0.60=0.24\), or \(24 \%\) chance of winning the election. Assume that each candidate can increase his probability of winning a state by \(10 \%\) if he, and not his opponent, visits that state. If both candidates visit the same state, there is no effect. a. Set up a payoff matrix with McCain as the row player and Obama as the column player, where the payoff for a specific set of circumstances is the probability (expressed as a percentage) that McCain will win both states. b. Where should each candidate visit under the circumstances?

A matrix is symmetric if it is equal to its transpose. Give an example of a. a nonzero \(2 \times 2\) symmetric matrix and b. a nonzero \(3 \times 3\) symmetric matrix.

Wrestling Tournaments City Community College (CCC) plans to host Midtown Military Academy (MMA) for a wrestling tournament. Each school has three wrestlers in the \(190 \mathrm{lb}\). weight class: \(\mathrm{CCC}\) has Pablo, Sal, and Edison, while MMA has Carlos, Marcus, and Noto. Pablo can beat Carlos and Marcus, Marcus can beat Edison and Sal, Noto can beat Edison, while the other combinations will result in an even match. Set up a payoff matrix, and use reduction by dominance to decide which wrestler each team should choose as their champion. Does one school have an advantage over the other?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure 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.