-1

A square matrix A is symmetric if $A^T = A$. 1. Give an example of a 3 × 3 symmetric matrix with entries which are all non-zero. 2. Prove that if $A^T*A = A$, then A is symmetric and $A = A^2$ .

user136733
  • 17
  • 3
  • What have you tried? Is this homework? Hint: show that for any $B$, $B^TB$ is symmetric: just calculate the transposed matrix. – André Caldas Apr 02 '14 at 13:48

1 Answers1

2

Hint: It's enough to prove that $A$ is symmetric. Just apply the definition to prove this.

Git Gud
  • 31,706
  • This could have been a comment... – André Caldas Apr 02 '14 at 13:45
  • @AndréCaldas comments and answers are distinguished based on content not length. Just because GitGud refuses to give a homework assignment away, doesn't mean it doesn't answer the question. – Guy Apr 02 '14 at 13:46
  • 1
    @AndréCaldas It could, I thought about it. But there is no middle ground between this and a full answer, so I opted for 'answering'. – Git Gud Apr 02 '14 at 13:46
  • @AndréCaldas I think this is a perfectly good answer. This is a homework problem most likely so we shouldn't do the entire problem for the student. Good hints are just as good (if not better) than full solutions. – Cameron L. Williams Apr 02 '14 at 13:47
  • @CameronWilliams I agree. If anything, full solutions to homework problems(unless the OP has clearly tried and failed) are actually bad. – Guy Apr 02 '14 at 13:47
  • @Sabyasachi: I feel you just want to give the guy a "lesson"... By the way, who is talking about length? I am talking about quality! – André Caldas Apr 02 '14 at 13:54
  • @CameronWilliams: You are under the impression I want you to give the OP the entire solution for his homework! Where did you get this information? – André Caldas Apr 02 '14 at 13:55
  • @AndréCaldas if by lesson you mean lesson in mathematics, then yes, that is the purpose of this site. Otherwise no. (Also how do you propose to improve quality without a full solution) – Guy Apr 02 '14 at 13:56
  • @Sabyasachi: "unless clearly tried and failed" -- Well, if this is not the case, you should point it out to him!!! Even a let me Google that for you would be better then this answer. – André Caldas Apr 02 '14 at 13:56
  • @AndréCaldas I meant that a full solution is only warranted in a case where the OP has shown an attempt to solve it, but the solution depends on some insight that hasn't "clicked" yet. In that case a full solution is ok. Btw this answer does teach. It shows how to deal with transposes in general. – Guy Apr 02 '14 at 13:58
  • @Sabyasachi: How to improve quality -- you can point out to the guy what he is NOT doing, for instance. You can give him definitions and call his attention to tricky parts of the definition, or whatever you think it might be hard to grasp for a beginner. I think there are lots of ways you can help the guy even if you think he is being lazy... – André Caldas Apr 02 '14 at 13:58
  • @Sabyasachi: I am happy we talked about the issue! Cheers... :-) – André Caldas Apr 02 '14 at 13:59
  • @AndréCaldas give him definitions:refer to the part of the answer where it basically says,this is straightforward. Apply the definition. Cheers. :) – Guy Apr 02 '14 at 14:00
  • @AndréCaldas you will notice that this is accepted, meaning it helped. – Guy Apr 02 '14 at 14:00
  • @Sabyasachi: It also got at least 2 "points". Meaning it has very good quality. Better then this, for example: http://math.stackexchange.com/questions/720415/inclusion-of-sigma-algebra-generated-by-random-variables?answertab=votes#tab-top – André Caldas Apr 02 '14 at 14:05
  • If you are being sarcastic, just state your point. – Guy Apr 02 '14 at 14:07
  • @Sabyasachi: Sorry, I was being sarcastic. My point is that you conclusion does not follow from your premises. But I will not comment further here. Cheers... – André Caldas Apr 02 '14 at 14:14