1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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
    Okay.
    A2=I+I2

    A=[tex]\sqrt{I+I^2}[/tex]

    2aA102=1

    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

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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
    A2=I

    2a(A2)51=1

    2a(I)51=1

    2a=(I)-51

    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

    Mark44

    Staff: Mentor

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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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: 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;

    (A101)-1=2aA

    [tex]\frac{1}{(A^2)^5^0}[/tex]=2a

    2I-50=2a
     
  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

    Mark44

    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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.

    (A+I)(A-I)=I

    A2-I2=I

    A2=2I

    (A2)51=(2I)51

    A102=251I51

    since I51=I

    (A102)-1=(251I)-1

    A-102=2-51I

    A-101xA-1=2-51I

    A-101=2-51IA

    A-101=2-51A

    since originally, A-101=2axA

    a must equal -51

    thanks for all of your input.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Matrix problem
  1. Matrix problem (Replies: 1)

  2. Matrix problem (Replies: 9)

  3. Matrix problem (Replies: 21)

  4. Matrix problem (Replies: 13)

  5. Matrix Problem (Replies: 5)

Loading...