# group matrix multiplication

• December 6th 2009, 04:02 PM
saambre
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
• December 6th 2009, 05:35 PM
lvleph
Can you use $\det(AB) = \det(A)\det(B) = 0 \Leftrightarrow \det(A) = 0 \text{ or } \det(B) = 0$?
• December 7th 2009, 12:59 AM
saambre
na
na we cant. We havent been taught that
• December 7th 2009, 01:07 AM
Swlabr
Quote:

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 $B^{-1}A^{-1}$...!
• December 9th 2009, 02:21 AM
saambre
thanks