1 of 1 If A is an n x n matrix, and x and y are different vectors in so that Ax=Ay, explain why this means that A cannot be invertible.Please remember to attach your solution as a .pdf file.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
Example: Jane Johnson – Written Homework Week #1
Do not copy the question text to your solution document. Just the answers including
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, don’t 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.
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
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.
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
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:
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
Examples of Good and Bad Solutions
Here is an example problem: if , are × matrices, = , ≠ , then must be
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
Examples of bad solutions:
1. = → −1 = −1 → =
This solution is missing the logic of what’s 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 = (
) and = (
), then = and ≠ . is singular.
This solution shows an example. It illustrates the general principle, but it does not prove
the general validity of the principle.
), = , ≠ . singular. This solution combines the
mistakes of the first and the second example. It shows no explanations, just an example,
and it doesn’t 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
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