/*! 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 a. Find values of the scalars \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 7

a. Find values of the scalars \(a\) and \(b\) so that $$ a\left[\begin{array}{rr} 12 & 3 \\ 0 & -9 \end{array}\right]+b\left[\begin{array}{rr} 4 & 1 \\ 0 & -3 \end{array}\right]=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] $$ b. Find values of the scalars \(a\) and \(b\) so that $$ a\left[\begin{array}{rr} 12 & 3 \\ 0 & -9 \end{array}\right]+b\left[\begin{array}{rr} 4 & 1 \\ 0 & -3 \end{array}\right]=\left[\begin{array}{rr} -28 & -7 \\ 0 & 21 \end{array}\right] $$ c. Are there solutions to the equations in parts a and b other than the ones you found?

Short Answer

Expert verified
a. There are infinitely many solutions; one such possible set could be \( a = 1 \) and \( b = -3 \). b. There are no solutions. c. For the first part, there are other solutions as any values of \( a \) and \( b \) satisfying \( b = -3a \) will work. For the second part, no different solutions can exist, as no solutions exist at all.

Step by step solution

01

Equate corresponding entries for first equation

Equate corresponding entries in the two matrices as follows:\n For the first entry, equate \( 12a + 4b = 0 \). For the second one, equate \( 3a + b = 0 \). For the third one, equate \( -9a - 3b = 0 \). Equating these entries gives us a system of linear equations.
02

Solve the system of linear equations for the first part

From the second equation, we find that \( b = -3a \). Plugging this into the first equation we get \( 12a - 12a = 0 \), which simplifies to \( 0 = 0 \). This indicates that the system has infinitely many solutions. Therefore, any real number can be a solution for \( a \) and \( b \) such that \( b = -3a \). For example, if we take \( a = 1 \), then \( b = -3 \). Similarly, if \( a = -2 \), \( b = 6 \) and so on. Any of these pairs of solutions will satisfy the given matrix equation.
03

Repeat the same process for the second matrix equation

For the second one, the system of equations become \( 12a + 4b = -28 \), \( 3a + b = -7 \) and \( -9a - 3b = 21 \). Solving this, we find that \( b = -3a \), just as with the first one. However, when we substitute \( b \) back into the first equation, we find that \( -24 = -28 \), which is not true. Therefore, there are no solutions for this second matrix equation.
04

Answer the third part of the problem

For the first part, there are infinitely many solutions because any pair of real numbers \( (a, b) \) such that \( b = -3a \) will work. Thus, the solutions that were found are not the only ones. However, for the second part, no solutions exist. The values derived for \( a \) and \( b \) in this fitness do not satisfy the given matrix equation.

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!

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

Give some useful interpretations for the vector space \(\mathbb{R}^{n}\) with \(n \geq 4\). What roles do the operations of addition and scalar multiplication play in your interpretations?

In Exercises 1 to 12, give step-by-step proofs of the identities stated in Theorem 1.7. Follow the format of the proofs given for part a and part e. If you are careful to take only one step at a time, it should be obvious what result you are using; hence, you will not need to write down the justification of each step. Prove Theorem 1.7, part \(c: \mathbf{v}-\mathbf{w}=-\mathbf{w}+\mathbf{v}\).

Talk to your friends about their views of a number such as 15. Do they think of it in completely abstract terms for its role in arithmetic and algebra, do they envision three rows of five unspecified objects, or do they think in concrete terms of 15 pencils or other specific objects? Does the size of the number influence the responses? Do math students tend to think about numbers differently than students in other fields of study? How aware are people of the level of abstraction with which they view numbers? Did they reach this level of abstraction with a sudden flash of insight or by working with numbers over a period of time?

Determine whether we obtain a vector space from \(\mathbb{R}^{2}\) with operations defined by $$ \begin{aligned} \left(v_{1}, v_{2}\right)+\left(w_{1}, w_{2}\right) &=\left(v_{1}+w_{1}, v_{2}+w_{2}\right), \\ r\left(v_{1}, v_{2}\right) &=(0,0) . \end{aligned} $$ It is especially interesting to have an example where exactly one of the axioms fails. This indicates that such an axiom is essential in the sense that there is no way to derive it from the other axioms.

Give some rules of ordinary real number arithmetic that do not have counterparts among the vector space axioms. Will any of these rules hold when interpreted in the context of a vector space?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.