Why can’t matrix A be invertible if Ax = Ay for different vectors x and y?

  

1 of 1 If A is an n x nmatrix, and x and y aredifferentvectors inso that Ax=Ay, explain why this means that A cannot be invertible.Please remember to attach your solutionas a .pdffile.
Written Homework Instructions for MAT 343
Each week, there is a written homework assignment that is to be submitted online, through
blackboard. Do not post your answer as a text submission. Submit it as a PDF file and use
“attach file” to submit it.
Your solution document must be a PDF file, not a doc, docx or ods, text field or a jpg. It must be
typed, not handwritten.
Your solution document must contain your name and the homework/week number as a
header on every page. Do not put in your student ID, class number, class time, or other
identifying information.
Example: Jane Johnson Written Homework Week #1
Do not copy the question text to your solution document. Just the answers including
supporting work.
If you violate the formatting guidelines in this document, you will lose points. In extreme cases,
the grader is permitted to refuse to grade your homework.
Important: you must show all work. If the question asks you to explain or similar, dont treat
that part as optional. It is the most important part. Explanations must be written in full,
grammatically correct sentences and should be succinct.
Naked equations (i.e. equations without verbal explanation) do not explain anything.
Explanations of a general principle must also be general, i.e. apply to all instances of the
situation under consideration. Showing an example does not prove a general principle.
Composing your written homework
You have several software options for composing your written homework.
Microsoft Word
By far the easiest option is to use Microsoft Word. You will have to use the equation editor to
create equations. It is found on the INSERT tab on the right side. The editor has a convenient
interface that lets you create fractions, radicals, vectors and matrices and other mathematical
symbols easily.
Open Office/Libre Office Writer
Open Office/Libre Office can create mathematical equations too, but the editor is rudimentary
and unintuitive, and the formula language lacks good documentation. The functionality is
shamefully hidden under Insert/Object/Formula. I do not recommend this option.
LaTeX
A free alternative to MS Word that produces superior results for scientific documents is LaTeX.
It is the standard software for serious scientific typesetting, but requires a greater learning
investment. If you are thinking about pursuing graduate studies, you will be doing yourself a
favor by learning and mastering LaTeX. It’s either now or later.
There is a good free LaTeX tutorial at
http://mirrors.ctan.org/info/lshort/english/lshortletter.pdf
Under Windows, the standard LaTeX distribution is MiKTeX: http://miktex.org/
There is a 3rd party editor for Windows called WinEdt which seamlessly integrates with MiKTeX
and greatly simplifies and streamlines the process of creating LaTeX documents:
http://www.winedt.com/
All these systems can export files to PDF format, the format in which you are required to
submit your solutions.
No matter what system you use, you are expected to produce proper mathematical notation.
Avoid runaway equations and use appropriate line breaks. Do not typeset exponential
expressions as a^b, fractions like (a+1)/(b+1) or roots like sqrt(..). Do not represent matrices in
MATLAB form.
Examples of Good and Bad Solutions
Here is an example problem: if , are matrices, = , , then must be
singular.
Example of a good solution:
If was regular, then we could left-multiply the equation = by 1 and obtain = .
That contradicts the assumption . Therefore, cannot be regular and must therefore be
singular.
Examples of bad solutions:
1. = 1 = 1 =
This solution is missing the logic of whats going on, namely that we explore the
consequence of being regular and show that it leads to a contradiction with one of the
premises. Your reasoning needs to be explained in full sentences, not just implied.
2. If we let = (
1 0
1 0
) and = (
), then = and . is singular.
0 0
0 0
This solution shows an example. It illustrates the general principle, but it does not prove
the general validity of the principle.
1 0
1 0
) =(
), = , . singular. This solution combines the
0 0
0 0
mistakes of the first and the second example. It shows no explanations, just an example,
and it doesnt even explain the example in complete sentences.
3. = (
4. If was regular, then we could left-multiply the equation = by ^-1 and obtain
= . That contradicts the assumption != I. Therefore, cannot be regular and must
therefore be singular.
While this solution is substantially correct, it does not use proper mathematical
typesetting.

Introduction:

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Written Homework Instructions for MAT 343 contains important guidelines that students need to follow when submitting their written homework assignments online. These instructions ensure that the submitted assignments meet the required standards and are easy for the graders to assess.

Description:

The written homework instructions for MAT 343 provide guidelines that students need to adhere to when submitting their homework assignments online. Students are required to submit their solutions in a PDF format and must not include any identifying information on their document, such as their student ID or class number. The document must contain their name and the homework week number on every page. The instructions also require students to show all their work and provide explanations where necessary. The explanations must be written in grammatically correct sentences and be succinct. Furthermore, the instructions provide students with software options, including Microsoft Word, Open Office/Libre Office Writer, and LaTeX, to use when composing their written homework assignments. These options come with different functionalities, and students are encouraged to choose the one that best suits their needs. The aim of the written homework instructions is to ensure that students submit well-formatted and high-quality homework assignments that are easy for the graders to assess.

Objectives:
– Understand the concept of invertible matrix in linear algebra.
– Apply the concept of matrix multiplication to solve problems related to matrix invertibility.
– Learn proper formatting and submission guidelines for written homework in MAT 343.

Learning Outcomes:
– Students will be able to determine if a matrix is invertible based on the result of matrix multiplication.
– Students will be able to format and submit their written homework assignments in MAT 343 correctly, adhering to the given guidelines.
– Students will be able to use equation editors (such as Microsoft Word or LaTeX) to properly format mathematical equations in written assignments.

Solution 1:

If Ax = Ay for different vectors x and y, it implies that A(x-y) = 0. Since x and y are different, (x-y) is not the zero vector. Therefore, A is not invertible since the existence of a non-zero vector in its null space means that A does not have a unique inverse.

Solution 2:

Suppose A is an invertible n x n matrix. Then, by the invertible matrix theorem, for any vector b in R^n, the system Ax = b has a unique solution. However, if Ax = Ay for different vectors x and y, then we have:

Ax = Ay

A(x-y) = 0

Since A is invertible, its null space is {0}. Hence, (x-y) = 0, which implies that x = y. This contradicts the assumption that x and y are different vectors. Therefore, if Ax = Ay for different vectors x and y, A cannot be invertible.

Suggested Resources/Books:
– “Linear Algebra and its Applications” by Gilbert Strang
– “Matrix Analysis and Applied Linear Algebra” by Carl D. Meyer
– “Introduction to Linear Algebra” by Gilbert Strang

Similar Asked Questions:
1. What is the importance of invertibility in linear algebra?
2. How do I determine if a matrix is invertible?
3. Can a non-square matrix be invertible?
4. What is the relationship between determinants and invertibility?
5. How can I use invertibility in solving systems of linear equations?

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