Proving multiplicative inverses of 2x2 matrix with elements in Z

[lostNfound]

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!

 

[Dick]

How is the determinant of the matrix related to the determinant of the inverse matrix?

 

[lostNfound]

If you're asking what the determinant of the inverse matrix is, then it is 1/(ad-bc).

 

[Dick]

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?