Let n be a positive integer.

(a) Let G be a set of real 2x2 matrices $A$ such that the $detA$ is a rational number of the form $m/n^t$ where $m$ and $t$ are nonnegative integers and $m\ne0$. Is G a group under multiplication?

(b) Same for $H$ of all real 2x2 matrices A such that $detA$ is of the form $n^t$ where $t\in \mathbb{Z}$?

I know the four defining properties of a group are closure, associativity, identity element, and inverse:

  1. Closure - For every pair of elements $a$ and $b$, $a*b$ must be an element of the group. Should I use $GL(2,{\Bbb{R}})$ as a mental reference (it's a group under multiplication) to see if the properties of a group hold for both $G$ and $H$ in same way?

  2. Associativity - For any three elements $a, b,$ and $c$, the equality $a*(b*c)=(a*b)*c$ must hold. I know multiplication of 2x2 matrices is always associative so this holds.

  3. Identity element - There is a group element $e$, the identity element, such that $a*e=e*a=a$ for any $a$ in the group.

  4. Inverse - For every group element $a$, there is an element $b$, the inverse, such that $a*b=b*a=e$, where $e$ is the identity. Looks like the determinants for $G$ and $H$ are not equal to $0$, so they have an inverse.

The determinant forms are throwing me off a bit...can someone provide some guidance? Or a more formal way of proving this?

  • 2
    $\begingroup$ Is $t$ a fixed integer? $\endgroup$ Nov 3, 2015 at 19:09
  • $\begingroup$ And what do you mean by using $GL(2,R)$ "as an example"? $\endgroup$ Nov 3, 2015 at 19:11
  • $\begingroup$ You cannot use just one example to prove that a property holds for all examples! $\endgroup$ Nov 3, 2015 at 19:16
  • $\begingroup$ The formal way to prove this is to start off with "Let A, B be in G. That means, by definition, that detA is [...] and detB is [...]. Then their product AB is also in G because its determinant is..." $\endgroup$ Nov 3, 2015 at 19:17
  • $\begingroup$ No no I understand that, I'm just saying would it be helpful to look at GL(2,R) as an example of a group under multiplication where these properties hold (just for my own reference)...I wasn't going to use this group to actually prove these necessary properties of G and H here. $\endgroup$
    – user0990
    Nov 3, 2015 at 19:20

1 Answer 1


Hint: In determining closure and inverses, note that $$ \det(AB) = \det(A)\det(B)\\ \det(A^{-1}) = 1/\det(A) $$

  • $\begingroup$ For G's closure, would showing that $\frac{mx}{n^ty^s}=\frac{m}{n^t}\frac{x}{y^s}$ where $x$ is nonnegative, $y$ and $s$ are positive integers suffice? $\endgroup$
    – user0990
    Nov 4, 2015 at 16:59
  • $\begingroup$ Also, can't we simply say that inversion holds for G since the $detA$ is nonzero? (Using the fact that an n × n matrix has an inverse if and only if its determinant is nonzero) $\endgroup$
    – user0990
    Nov 4, 2015 at 17:36
  • $\begingroup$ Just because a matrix in $G$ has an inverse, doesn't mean that inverse is itself an element of $G$ $\endgroup$ Nov 4, 2015 at 17:54
  • $\begingroup$ For closure: what you need to check is whether that product can itself be written in the form $m/n^t$ for some choice of integers $m,n,t$ $\endgroup$ Nov 4, 2015 at 17:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.