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Math Help - group matrix multiplication

  1. #1
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    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
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  2. #2
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    Can you use \det(AB) = \det(A)\det(B) = 0 \Leftrightarrow \det(A) = 0 \text{ or } \det(B) = 0 ?
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  3. #3
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    na

    na we cant. We havent been taught that
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by saambre View Post
    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}...!
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  5. #5
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    thanks

    thanks for your reply. was very helpful. cheers
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