Proving multiplicative inverses of 2x2 matrix with elements in Z

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The discussion revolves around the properties of 2x2 matrices with integer elements, specifically focusing on the conditions under which these matrices possess multiplicative inverses. The original poster is exploring the relationship between the determinant of such matrices and their ability to have inverses, suggesting that only those with determinants of +/- 1 qualify.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to establish a proof regarding the necessity of the determinant being +/- 1 for the existence of multiplicative inverses in 2x2 matrices with integer elements. Other participants inquire about the relationship between the determinant of a matrix and its inverse, with one noting that the determinant of the inverse is 1/(ad-bc).

Discussion Status

The discussion is ongoing, with participants raising questions about the properties of determinants and their implications for matrix inverses. There is no explicit consensus yet, but the exploration of the determinant's role in determining the existence of inverses is being actively pursued.

Contextual Notes

Participants are examining the implications of integer elements in the context of matrix inverses and determinants, questioning whether the determinant of the inverse can also be an integer.

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!
 
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How is the determinant of the matrix related to the determinant of the inverse matrix?
 
If you're asking what the determinant of the inverse matrix is, then it is 1/(ad-bc).
 
lostNfound said:
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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