1. Limited time only! Sign up for a free 30min personal 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!

Inverse of a matrix with an X

  1. Oct 21, 2009 #1
    Hi everyone!

    1. The problem statement, all variables and given/known data
    Find the inverse of the following using row reduction. If it does not exist, indicate clearly why.

    3, 0, 6;
    1, -2, x;
    1, 2, 1;

    2. The attempt at a solution

    I started by augmenting with a 3 by 3 Identity matrix:
    3, 0, 6, 1, 0, 0;
    1, -2, x, 0, 1, 0;
    1, 2, 1, 0, 0, 1;

    Then I started using Gaussian Elimination until I hit a road bump:

    3, 0, 6, 1, 0, 0;
    0, -2, x-3, 1/3, 1, 0;
    0, 0, x-5, 0, 0, 1;

    Should I just keep on pushing though? Or should I try to figure X out? Or is there some sort of theorem out there than can help me?
     
  2. jcsd
  3. Oct 21, 2009 #2

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Keep pushing through. What conditions would make the "pushing through" invalid?
     
  4. Oct 21, 2009 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    I'm going to guess you meant to start by taking (1/3) times the first row and subtracting it from the second row. Something has gone wrong already.
     
  5. Oct 21, 2009 #4
    Ok, I reworked it and I got this:
    3, 0, 6, 1, 0, 0;
    0, -2, 1, 1/3, 0, -1;
    0, 0, (X-3), -2/3, 1, 1;

    So would this matrix not have an inverse since I can never get the Identity on the left? It seems like it I'll keep getting number when I do row reduction after this...
     
  6. Oct 21, 2009 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    It's still wrong. Can you break this out one step at a time and say what you are doing? What happened to the x in the second row?
     
  7. Oct 21, 2009 #6

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Are you saying that x-3 does not have a multiplicative inverse?
     
  8. Oct 21, 2009 #7

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    She swapped the second and third rows.
     
  9. Oct 21, 2009 #8
    Here is my process:
    3, 0, 6, 1, 0, 0;
    1, -2, x, 0, 1, 0;
    1, 2, 1, 0, 0, 1;

    I multiplied -1/3 to r1 and added to r2 and got
    3, 0, 6, 1, 0, 0;
    0, -2, x-2, -1/3, 1, 0;
    1, 2, 1, 0, 0, 1;

    I then did the same and added to r3 and got
    3, 0, 6, 1, 0, 0;
    0, -2, x-2, -1/3, 1, 0;
    0, 2, -1, -1/3, 0, 1;

    I then added r2 to r3 and got
    3, 0, 6, 1, 0, 0;
    0, -2, x-2, -1/3, 1, 0;
    0, 0, x-3, -2/3, 1, 1;

    Then I added -r3 to r2 to get:
    3, 0, 6, 1, 0, 0;
    0, -2, 1, 1/3, 0, -1;
    0, 0, x-3, -2/3, 1, 1;

    So I got rid of the x in this step. Should I declare this not invertible?

    EDIT: It's He by the way, lol
     
  10. Oct 21, 2009 #9

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    No.

    What is the multiplicative inverse of x-3?
    Under what conditions does x-3 not have a multiplicative inverse?
     
  11. Oct 21, 2009 #10
    The multiplicative inverse is 1/(x-3), this does not apply when x=3...
    Hmm... I was thinking about this before but I felt it wasn't right. Should I multiply row 3 by the multiplicative inverse ?
     
  12. Oct 21, 2009 #11

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Oh, yeah. I see it now. Continue with your excellent help. And thanks for the exposition rey242.
     
  13. Oct 21, 2009 #12

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Give that man a prize!

    With that you will have the third row in the desired form. The rest is just tedious elementary algebra. It's easy to make a mistake in all this tedious work. I suggest that you check your work at the end by multiplying the original matrix and its supposed inverse to make sure it really is the inverse.
     
  14. Oct 21, 2009 #13
    Thanks for all your help

    I understand now.

    I really appreciate it!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Inverse of a matrix with an X
  1. Inverse matrix (Replies: 3)

  2. Matrix inversion (Replies: 6)

  3. Inverse of a Matrix (Replies: 2)

  4. Matrix inverse? (Replies: 2)

  5. Inverse matrix (Replies: 6)

Loading...