Linear Equation and Matrices

  • Thread starter hkus10
  • Start date
  • #1
hkus10
50
0
1) Let A be an n x n matrix. Prove that if Ax= 0 for all n x 1matrices, then A=O.
Can you show me the steps of solving this problem?
Please!
 

Answers and Replies

  • #2
36,195
8,182
1) Let A be an n x n matrix. Prove that if Ax= 0 for all n x 1matrices, then A=O.
Can you show me the steps of solving this problem?
Please!
Show us what you've tried. The rules of this forum say that we aren't supposed to provide any help if you haven't given the problem a try.
 
  • #3
hkus10
50
0
Show us what you've tried. The rules of this forum say that we aren't supposed to provide any help if you haven't given the problem a try.

Actually, I am stuck how to begin this question. However, I still have some ideas for how to start. I just want to make sure this is the right approach before solving this kind of proof problem. Is it related to the Matrix-Vector Product ?
 
  • #4
spamiam
360
0
A vector and an n x 1 matrix are the same thing: it's just semantics.

First, write down what it means for A to be the zero matrix. This will tell you what it is you need to prove. Next, the condition Ax = 0 holds for all n x 1 matrices (or vectors) x, so try to cleverly choose a vector that makes the product Ax nice to work with.

I'm being pretty vague, but I'll be happy to elaborate once you show us your attempt.
 
  • #5
hkus10
50
0
Is it I have to converse of the implication first?
 
  • #6
spamiam
360
0
I suppose you could, but I think it's easier to prove directly. I guess both methods boil down to the same thing in the end.

Try this: how would you prove that the entry [itex]A_{1,1}=0[/itex]? If you can do this, you should be able to generalize your method.
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
43,010
969
Suppose some entry in A, say Aij is NOT 0. Can you construct a vector v, such that Av is not 0?
 

Suggested for: Linear Equation and Matrices

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
936
  • Last Post
Replies
1
Views
960
  • Last Post
Replies
1
Views
999
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
1K
Replies
14
Views
2K
  • Last Post
Replies
4
Views
712
Replies
5
Views
3K
Top