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Linear algebra

  1. Mar 28, 2007 #1

    daniel_i_l

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    Gold Member

    1. The problem statement, all variables and given/known data
    a) find all the 2x2 matrices where AA=A.
    b) prove that if
    [tex]
    I + a_{1}A + a_{2}A^2 + ... + a_{k}A^k = 0
    [/tex]
    then A is invertable


    2. Relevant equations
    1)det(A) = 0 iff A isn't invertable


    3. The attempt at a solution
    a) I'm not sure how to approch this. I found that if A is invertable then the only solution is A=I but how do i cover the other cases?

    b) by rearanging:
    [tex]
    A(a_{1}I + a_{2}A^1 + ... + a_{k}A^{k-1}) = -I
    [/tex]
    and if i take the determinant on each side i see that |A| <> 0 so it's invertable. Is that correct?
    Thanks.
     
    Last edited: Mar 28, 2007
  2. jcsd
  3. Mar 28, 2007 #2

    radou

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    Homework Helper

    b) looks correct.

    Regarding a), assume A is regular and see what follows from that. Further on, consider A = I as a special case.
     
  4. Mar 28, 2007 #3

    Dick

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    AA=A says that A is a projection operator. So consider what the range of A is. The 'other' case you are after is where it is a one-dimensional subspace of R^2. BTW, you don't have to show det(A) is non-zero, you've constructed an explicit inverse.
     
  5. Mar 28, 2007 #4

    Dick

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    You could also just set A=[[a,b],[c,d]] and write the condition A^2=A. You can then eliminate two of the variables (except for singular cases). There are a two parameter family of these babies.
     
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