Proving multiplicative inverses of 2x2 matrix with elements in Z

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  • #1
lostNfound
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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!
 

Answers and Replies

  • #2
Dick
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How is the determinant of the matrix related to the determinant of the inverse matrix?
 
  • #3
lostNfound
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If you're asking what the determinant of the inverse matrix is, then it is 1/(ad-bc).
 
  • #4
Dick
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If you're asking what the determinant of the inverse matrix is, then it is 1/(ad-bc).

Right. Is the determinant of the inverse matrix an integer?
 

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