# Thread: group matrix multiplication

1. ## group matrix multiplication

[B]hi, im new to the forum

how do you show that GL(2,R) is a group under matrix mulitplication?

and then show that F(matrix)= [ {a b} {-b a}
, a,b are elements of R, and a,b dont equal 0]

thanks

2. Can you use $\det(AB) = \det(A)\det(B) = 0 \Leftrightarrow \det(A) = 0 \text{ or } \det(B) = 0$?

3. ## na

na we cant. We havent been taught that

4. Originally Posted by saambre
na we cant. We havent been taught that
You have four things to show:

Closure: $det(A) \neq 0$, $det(B) \neq 0$. 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. $(AB)C = \ldots = A(BC)$.

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 $det(A) \neq 0 \Longleftrightarrow A \text{ has an inverse}$. So, we can construct an inverse for $AB$. It is just $B^{-1}A^{-1}$...!