# Find an appropriate matrix according to specific conditions

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1. Nov 23, 2016

### 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.

3. The attempt at a solution

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

Thanks.

2. Nov 23, 2016

### Staff: Mentor

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

3. Nov 23, 2016

### Avibu

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

4. Nov 23, 2016

### Staff: Mentor

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

5. Nov 23, 2016

### 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.

6. Nov 23, 2016

### Staff: Mentor

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

7. Nov 23, 2016

### 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

8. Nov 23, 2016