Linear Algebra - Determinant Properties

  1. 1. The problem statement, all variables and given/known data

    1. Give an example of a 2x2 real matrix A such that A^2 = -I
    2. Prove that there is no real 3x3 matrix A with A^2 = -I

    2. Relevant equations

    I think these equations would apply here?
    det(A^x) = (detA)^x
    det(kA) = (k^n)detA (A being an nxn matrix)
    det(I) = 1

    3. The attempt at a solution

    1.
    Would I use above equations with this question? This is what I did so far; I don't know if I'm off in answering this question...
    I wrote:
    It is a 2x2 matrix, so n = 2
    det(A^2) = det(-I)
    (detA)^2 = (-1^2)detI
    (detA)^2 = detI (and detI = 1)

    Therefore, detA * detA must = 1; so could I use the identity matrix itself as a matrix example for A:
    A =
    [1 0
    0 1]
    Then, detA * detA = 1 = detI
    Does this make sense? Or am I not allowed to use the identity matrix here?

    2.
    I wrote:
    It is a 3x3 matrix, so n = 3
    det(A^2) = det(-I)
    (detA)^2 = (-1^3)detI
    (detA)^2 = -(detI )
    (detA)^2 = -1

    Then, can I just say that since (detA)^2 is always positive since it is squared... therefore, (detA)^2 can never equal -1, and there is no real 3x3 matrix A with A^2 = -I


    Thanks a lot for the help!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. gabbagabbahey

    gabbagabbahey 5,013
    Homework Helper
    Gold Member

    Your solutions look great to me! :smile:
     
  4. HallsofIvy

    HallsofIvy 40,930
    Staff Emeritus
    Science Advisor

    I hate to disagree with gabbagabbahey, but you haven't answered the questions at all!

    Have you forgotten what the question asked? You were asked to find A such that A2= -I. The example you give has A2= I, not -I.

    Yes, this part is correct.


     
  5. gabbagabbahey

    gabbagabbahey 5,013
    Homework Helper
    Gold Member

    Sorry, I had a small brain fart there :redface:...As Halls said, you are looking for an example of a matrix such that [itex]A^2=-I[/itex].
     
  6. Yeah, thanks! Oops!

    Well, since I am looking for an example of a matrix A where A^2 = -I, I could use the example A =
    [0 -1
    1 0]
    Since this squared = -I

    But, I found this using trial and error. Is there any other way to answer this question...? It seems the question is too easy if it's just asking for an answer... is there some sort of formula or something I can use here?

    Thanks again.
     
  7. Dick

    Dick 25,893
    Science Advisor
    Homework Helper

    You could think geometrically. The linear operator -I represents rotation by 180 degrees in the plane. The square root of that just might be rotation by 90 degrees, right? What's the matrix for that?
     
  8. Okay, so I could use properties from linear transformations and say:
    [0 -1
    -1 0]
    = -I, which equals:
    [cos180 -sin180
    sin180 cos180]

    Then, since A^2 =
    [cos180 -sin180
    sin180 cos180]

    I need the square root of this, which is:
    [cos90 -sin90
    sin90 cos90]

    So then A =
    [0 -1
    1 0]

    This may sound like a stupid question, but how do I take the square root of
    [cos180 -sin180
    sin180 cos180]
    to obtain:
    [cos90 -sin90
    sin90 cos90]

    How do you take the square root of sin180, cos180 etc... Sorry I haven't done math before linear algebra in 4 years... I can't remember this stuff.

    Thanks!!
     
  9. Dick

    Dick 25,893
    Science Advisor
    Homework Helper

    A^2(x)=A(A(x)). It just means, 'do A twice'. If you rotate by 90 twice, you get a rotation by 180. That's why I said think geometrically. Of course, you could also rotate by -90. There are two 'square roots'.
     
  10. Ok got it. Thanks for the help!
     
  11. HallsofIvy

    HallsofIvy 40,930
    Staff Emeritus
    Science Advisor

    Or write [tex]A= \left[\begin{array}{cc}a & b \\ c & d\end{array}\right][/tex] so that [tex]A^2= \left[\begin{array}{cc}a^2+ bc & ab+ bd \\ ac+ cd & bc+ d^2\end{array}\right]= \left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right][/tex]

    So we have [itex]a^2+ bc= -1[/itex], [itex]ab+ bd= 0[/itex], [itex]ac+ cd= 0[/itex], [itex]bc+ d^2= -1[/itex], four equations to solve for a, b, c, and d.
     
  12. Please help me with these:
    1) Prove that: nxn real matrix A is a root of f(X)= a[n].X^n+...+a[0].I, where a[n],...,a[0] are coefficients of the polynomial P(t)= det [A-t.I]
    2) Let 5x5 real matrix A be satisfied: A^2008 = 0. Prove that: A^5=0.
    Thanks.
     
  13. gabbagabbahey

    gabbagabbahey 5,013
    Homework Helper
    Gold Member

    Please start a new thread for your question and show an attempt. We don't know how much help you need and in which areas if you don't show us what you've tried or explain where you are having issues with the problem.
     
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