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!

Proving multiplicative inverses of 2x2 matrix with elements in Z

  1. Nov 15, 2011 #1
    So in describing the elements of M2(Z) that have multiplicative inverses, the answer that I keep coming back to is that the only ones are those with determinants of +/- 1, because the determinant would have to be able to divide all elements. I think I've conifrmed this scouring the web, but nobody has actually proved it. They just say that those elements of M2(Z) with multiplicative inverses are those with determinant +/-1, with no formal proof.

    I know if you have matrix [a,b; c,d], the inverse is [d/(ad-bc), -b/(ad-bc); -c/(ad-bc), a/(ad-bc)], with ad-bc being the determinant. I'm wondering if anyone can prove that this determinant must be equal to 1 or -1 in order for the elements of M2(Z) to have multiplicative inverse.

    Thanks!
     
  2. jcsd
  3. Nov 15, 2011 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    How is the determinant of the matrix related to the determinant of the inverse matrix?
     
  4. Nov 15, 2011 #3
    If you're asking what the determinant of the inverse matrix is, then it is 1/(ad-bc).
     
  5. Nov 15, 2011 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Right. Is the determinant of the inverse matrix an integer?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proving multiplicative inverses of 2x2 matrix with elements in Z
Loading...