Proving multiplicative inverses of 2x2 matrix with elements in Z

In summary, the only elements in M2(Z) with multiplicative inverses are those with determinants of +/- 1, which can be proven by the fact that the determinant of the inverse matrix is equal to 1/(ad-bc). This determinant must be an integer for the elements in M2(Z) to have multiplicative inverses.
  • #1
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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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  • #2
How is the determinant of the matrix related to the determinant of the inverse matrix?
 
  • #3
If you're asking what the determinant of the inverse matrix is, then it is 1/(ad-bc).
 
  • #4
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?
 

1. How do you determine if a 2x2 matrix has a multiplicative inverse with elements in Z?

To determine if a 2x2 matrix has a multiplicative inverse in Z, you need to calculate the determinant of the matrix. If the determinant is nonzero and relatively prime to the modulus of Z (usually denoted as p), then the matrix has a multiplicative inverse in Z.

2. What is the formula for finding the multiplicative inverse of a 2x2 matrix with elements in Z?

The formula for finding the multiplicative inverse of a 2x2 matrix with elements in Z is:

1/det(A) * adj(A) mod p, where A is the original matrix, det(A) is the determinant, adj(A) is the adjugate matrix, and p is the modulus of Z.

3. Can a 2x2 matrix have a multiplicative inverse in Z if its determinant is zero?

No, a 2x2 matrix cannot have a multiplicative inverse in Z if its determinant is zero. This is because the multiplicative inverse requires dividing by the determinant, which is not possible if the determinant is zero.

4. Are there any restrictions on the elements of a 2x2 matrix for it to have a multiplicative inverse in Z?

Yes, the elements of a 2x2 matrix must be integers and relatively prime to the modulus of Z in order for it to have a multiplicative inverse in Z. If the elements do not meet these requirements, then the matrix will not have a multiplicative inverse.

5. Can a 2x2 matrix have more than one multiplicative inverse in Z?

No, a 2x2 matrix can only have one multiplicative inverse in Z. This is because the inverse is unique and there cannot be multiple values that satisfy the inverse calculation.

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