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Find an appropriate matrix according to specific conditions

  1. Nov 23, 2016 #1
    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:

    • 396Ar.png is an eigenvector of A with eigenvalue of 10 , and izSFL.png is an eigenvector of A with eigenvalue of 20
    .

    • 396Ar.png 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 :

    SL6DX.png



    Can you please assist ?
    Thanks.
     
  2. jcsd
  3. Nov 23, 2016 #2

    DrClaude

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    Staff: Mentor

    What is the relation between ##(1, 3)^T## and ##(2, 6)^T##?
     
  4. Nov 23, 2016 #3
    I am not sure if I understood your question but the vectors seem to be linearly dependent
     
  5. Nov 23, 2016 #4

    DrClaude

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    Staff: Mentor

    Exactly. So what happens when you apply ##A## to those vectors?
     
  6. Nov 23, 2016 #5
    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.
     
  7. Nov 23, 2016 #6

    DrClaude

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    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##?
     
  8. Nov 23, 2016 #7
    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
     
  9. Nov 23, 2016 #8

    DrClaude

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    Staff: Mentor

    Isn't this a contradiction?
     
  10. Nov 23, 2016 #9
    It is ! :)
    It means there are no appropriate matrices.
    Thank you so much for your time and your assitance!
     
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