Find an appropriate matrix according to specific conditions

[Avibu]

I am facing some difficulties solving one of the questions we had in our previous exam. I am sorry for the bad translation , I hope this is clear.

In each section, find all approppriate matrices 2x2 (if exists) , which implementing the given conditions:

• 
  is an eigenvector of A with eigenvalue of 10 , and
  
  is an eigenvector of A with eigenvalue of 20

.

• 
  is an eigenvector of A with eigenvalue of 10 , and EXISTS eigenvector of A with eigenvalue of 20

If there are no matrices matched , explain why.

The Attempt at a Solution

[/B]
I tried to build equations for the first section but I have no idea how to keep from there :

Can you please assist ?
Thanks.

 

[DrClaude]

What is the relation between $(1, 3)^T$ and $(2, 6)^T$?

 

[Avibu]

I am not sure if I understood your question but the vectors seem to be linearly dependent

 

[DrClaude]

I am not sure if I understood your question but the vectors seem to be linearly dependent

Exactly. So what happens when you apply $A$ to those vectors?

 

[Avibu]

If I put those equations from step 3 (above) in a matrix , I will have 2 rows filled with 0's and Rank A < Rank(A|b) => No solution?
I would apprciate if you could explain it better than I do ,becasue I really want to understand what I am doing and how it should be solved.

 

[DrClaude]

Keep it simpler. You have a vector $v$ such that
$$
A v = 10 v
$$
Take a second vector $u = 2 v$. What is $Au$?

 

[Avibu]

Ok so, u=2v

{ Av = 10v
{ Au = 20u

{ Av = 10v
{ A(2v) = 20(2v)

{ Av = 10v
{ 2(Av) = 40v

{ Av = 10v
{ Av = 20v

I think I missed the point , I can see what you are trying to do but still can't figure it out

 

[DrClaude]

Ok so, u=2v
{ Av = 10v
{ Av = 20v

Isn't this a contradiction?

 

[Avibu]

It is ! :)
It means there are no appropriate matrices.
Thank you so much for your time and your assitance!