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Homework Help: Matrix problem

  1. Oct 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Find "a" when A is a square matrix satisfying (A+I)(A-I)=I and (A101)-1=2axA

    I is the identity matrix.

    3. The attempt at a solution
    I'm trying to find A. I didn't know where to begin, so I picked A to be all zeroes and plugged it in the equation. It didn't work...
    I tried A =
    -1 -1
    -1 -1
    I ended up with
    1 2
    2 1

    I want
    1 0
    0 1

    Can some one give me a hint please.
  2. jcsd
  3. Oct 24, 2009 #2


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    The point isn't to find the matrix A, it's to find the number 'a'. Expand the first equation and learn something about A^2. Multiply both sides of the second equation by A^(101). Hmm?
  4. Oct 24, 2009 #3



    I don't see a substitution that will help there.

    I didn't know I could expand the first equation.

    I'm not sure if I even multiplied through by A101 correctly.
  5. Oct 24, 2009 #4


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    You've got A^2=2I since I^2=I. Don't bother with the sqrt, you don't need to find A and you can't do it that way anyway. A^(102)=(A^2)^51. Now do you see it?
  6. Oct 24, 2009 #5




    a ln 2=-51 ln I

    a= -51(ln I/ ln 2)

    a= -51 ln (I-2)

    Is that close?
  7. Oct 25, 2009 #6


    Staff: Mentor

    (above) No, A2 = 2I.
    Should be I, not 1, on the right side.
  8. Oct 25, 2009 #7


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    Also [itex](A^2)^{51}= A^{102}[/itex], not [itex]A^{101}[/itex]

    And note that you want [itex]A^{-101}[/itex].

    Knowing that [itex]A^2= 2I[/itex],what is [itex]A^{-2}[/itex]?

    It also helps to know that 101= 2(50)+ 1.
    Last edited by a moderator: Oct 25, 2009
  9. Oct 25, 2009 #8
    I don't see how it is supposed to be an I on the right side instead of 1.
    If I have (A101)-1 thats just 1/A101.
    If I multiply through by A101 then don't I have a 1 on the right side?

    I tried this;



  10. Oct 25, 2009 #9
    Because both A and I are matrices. You mismatch the elements if you set it equal to 1.

    No, becuase A*A-1 = I, not 1.

    Remember, as was pointed out before, A2 = 2*I and A101=(A2)50*A
  11. Oct 25, 2009 #10


    Staff: Mentor

    No, no, no! Matrix division is not defined!
    Multiply both sides of the equation above by A101.
    What is A101(A101)-1?
    Edit: Moved a right parenthesis.
    What is A1012aA?
    What can you replace A2 with?
    As already noted, you can't divide by a matrix.
    Last edited: Oct 25, 2009
  12. Oct 25, 2009 #11


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    You can't define matrix "division" as "multiply by [itex]A^{-1}[/itex]" for two reasons: 1) Many matrices do not have inverses.

    2) If A does have an inverse, multiplying on left or right will typically give different results.
  13. Oct 25, 2009 #12
    Okay, I got it.






    since I51=I






    since originally, A-101=2axA

    a must equal -51

    thanks for all of your input.
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