Simple Linear Algebra (determinant invertibility)

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Discussion Overview

The discussion revolves around the invertibility of a square matrix A given the condition that det(A^4) = 0. Participants explore the implications of this condition and the reasoning behind the conclusion that A cannot be invertible, while also addressing the nature of mathematical proofs in general.

Discussion Character

  • Technical explanation
  • Homework-related
  • Debate/contested

Main Points Raised

  • One participant presents a reasoning process showing that if det(A^4) = 0, then det(A) must also be 0, leading to the conclusion that A cannot be invertible.
  • Another participant agrees with the initial reasoning and emphasizes that there is not a single "correct way" to approach the proof, suggesting that multiple valid proofs exist.
  • There is a reminder about the importance of demonstrating logical correctness in proofs, as well as the subjective nature of elegance in mathematical arguments.
  • Several posts diverge from the main topic, discussing unrelated requests for book recommendations on group theory and proper posting etiquette in the forum.

Areas of Agreement / Disagreement

While there is agreement on the correctness of the initial reasoning regarding the invertibility of matrix A, the discussion also includes unrelated topics, indicating a lack of consensus on the focus of the thread.

Contextual Notes

Some participants express uncertainty about whether additional work is needed to demonstrate understanding of the proof, highlighting the subjective nature of what constitutes a complete proof in this context.

slashrulez
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I have a quick question about whether or not a matrix is invertible. The question asked is pretty simple, "Suppose that A is a square matrix such that det(A^4) = 0. Show that A cannot be invertible." I know how to explain it, but I'm not sure if it's really the "correct" way, as in I'm not missing anything or making assumptions I wouldn't be allowed to make on an exam.

So my stab at it:

det(A^4) = det(AAAA) = detA detA detA detA = 0, therefore det A = 0

If A is invertible, there exists a B such that

AB = BA = I
det(AB) = det(I)
detA detB = 1
Therefore, for the matrix to be invertible, detA must be non-zero, which it isn't

Like, that seems right to me, but I'm not sure if I have to do any additional work for the part with the inverse to show I understand it.
 
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Looks perfect to me.
 
slashrulez said:
I have a quick question about whether or not a matrix is invertible. The question asked is pretty simple, "Suppose that A is a square matrix such that det(A^4) = 0. Show that A cannot be invertible." I know how to explain it, but I'm not sure if it's really the "correct" way, as in I'm not missing anything or making assumptions I wouldn't be allowed to make on an exam.

So my stab at it:

det(A^4) = det(AAAA) = detA detA detA detA = 0, therefore det A = 0

If A is invertible, there exists a B such that

AB = BA = I
det(AB) = det(I)
detA detB = 1
Therefore, for the matrix to be invertible, detA must be non-zero, which it isn't

Like, that seems right to me, but I'm not sure if I have to do any additional work for the part with the inverse to show I understand it.

The only thing I would add to the earlier approval is that there is not a single "correct way". You logic is flawless and the conclusion correct, so it is at the least a correct way.

The only real rule for doing a proof is that the logic be correct and that you demonstrate that which was to be demonstrated. There are often many valid proofs of a single theorem. Some are perhaps more elegant than others, but elegance is in the eye of the beholder and quite often the elegant proofs result from reviewing and refining that which was discovered by others.
 
cdniki said:
Hi
Can anybody suggest me a good book for algebra(group theory)which can be helpful me for math entrance exam.

I am not quite sure why you would be looking for a book on group theory for an entrance exam. However, Mashall Hall's The Theory of Groups is a classic on that subject and quite good. Rotman's An Introduction to the Theory of Groups is also good.
 
If you're posting a homework problem you should make a new thread in the homework section. Read the sticky at the top of the section seeing what you should include (in particular, you should include work that you've done)
 
cdniki said:
Hi
Can anybody suggest me a good book for algebra(group theory)which can be helpful me for math entrance exam.
A good introductory book is "A First Course in Abstract Algebra". ISBN 0-201-76390-7
 
cdniki's posts were removed as he needs to post in the proper forum and show his work.
 

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