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Property of Determinants Answers Check

  1. May 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Let A and P be square matrices of the same size with P invertible, Prove detA=det(P-1AP)

    2. Relevant equations
    Suppose that A and B are square matrices of the same size. Then det(AB)=det(A)det(B)

    3. The attempt at a solution

    detA=det(P-1AP)
    detA=det(P-1PA)
    detA=det(IA)
    detA=1*detA
    detA=detA

    SECOND QUESTION:

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

    Let A be an nxn matrix. Prove that if matrix A satisfies 7A2+8A+3I=[0]

    2. Relevant equations

    Invertible Matrix Theorem

    3. The attempt at a solution

    7A2+8A+3I=[0]
    7A2+8A = -3I
    A(7A+8)=-3I

    From this point I dont know if im headed towards the correct answer or have the right idea


    THIRD QUESTION:

    1. The problem statement, all variables and given/known data
    Suppose A is an nxn matrix satisfying AT+A=[0], where n is odd. Prove detA=0.

    2. Relevant equations
    detAT=detA

    3. The attempt at a solution

    AT+A=[0]
    AT=-A
    detAT=det(-A)
    since detAT=detA
    detA=det(-A)

    I think i've got the right idea...
     
    Last edited: May 9, 2013
  2. jcsd
  3. May 9, 2013 #2

    Mark44

    Staff: Mentor

    You can't do this (above). Matrix multiplication is not generally commutative, so in general, AP ≠ PA
    Typo in your equation?
    Typo?
    How does this show that det(A) = 0?
    Also, what role does the fact that n is odd play?
     
  4. May 9, 2013 #3
    sorry for the typos I edited and fixed them
     
  5. May 9, 2013 #4
    that is where i am stumped... is it det(A) = -det(A) then detA=0 ?
     
  6. May 9, 2013 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    IF det(A)=(-det(A)) then det(A)=0, sure. det(-A) isn't always equal to -det(A). What is det(kA) for k a constant?
     
  7. May 9, 2013 #6
    kndet(A)
     
  8. May 9, 2013 #7

    Dick

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    Science Advisor
    Homework Helper

    Well, sure. Now put k=(-1) and figure out what n being odd might have to do with it.
     
  9. May 9, 2013 #8
    det(AT)=det(-A)
    det(A)=(-1)ndetA

    from here i still don't know how det=0
     
  10. May 9, 2013 #9

    Dick

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    Science Advisor
    Homework Helper

    It only has to be zero if n is odd. Then (-1)^n=(-1), right? det is a real number. If a real number x=(-x) then x=0. Can you prove that?
     
  11. May 9, 2013 #10
    does it have anything to do with additive inverse?
     
  12. May 9, 2013 #11

    Mark44

    Staff: Mentor

    Don't overthink this.

    Since n is odd, (-1)n = -1, so
    det(AT) = det(-A) = (-1)ndet(A) = ?
    What can you conclude from the above?
     
  13. May 9, 2013 #12

    Mark44

    Staff: Mentor

    Also, it's better to post one problem per thread.
     
  14. May 9, 2013 #13

    Mark44

    Staff: Mentor

    The last line should be A(7A + 8I) = -3I
    It doesn't make sense to add a number to a matrix.

    Rearranging a bit results in this equation.
    So A * (-1/3)(7A + 8I) = I

    Does this give you any ideas?
     
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