Can you use ?
You have four things to show:
Closure: , . We shall do this last of all...
Associativity: You know this holds from your linear algebra course (there is nothing special about multiplication in GL, it is just the same as normal matrix multiplication which is associative), but there is nothing wrong with going through it one more time. .
Identity: You know what this is...
Inverses: Again, you know what the inverse of a matrix with non-zero determinant is. Just look up your linear algebra notes!
So, we still need to prove closure. Notice that . So, we can construct an inverse for . It is just ...!