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Homework Help: Isomorphism: matrix determinant

  1. Jan 19, 2009 #1
    Determine whether the given map [tex]\varphi[/tex] is an isomorphism of the first binary structure with the second.
    < M2(R ), usual multiplication > with <R, usual multiplication> where [tex]\varphi[/tex](A) is the determinant of matrix A.

    The determinant of the matrix is ad-bc, so [tex]\varphi[/tex](A)=ad-bc.
    For this to be an isomorphism, I have to show that the function is one to one, onto and preserves the operations.
    I'm having trouble getting this to work. Any suggestions?
     
  2. jcsd
  3. Jan 19, 2009 #2

    Hurkyl

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    Maybe it can't....
     
  4. Jan 19, 2009 #3
    According to the book it is suppose to be an isomorphism, the question says it is. I can get it to be one to one and onto, but i am having trouble with it preserving the operations.
     
  5. Jan 19, 2009 #4

    HallsofIvy

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    You just said that you must prove the function is "one to one". That is that two different matrices, such as
    [tex]\begin{bmatrix}2 & 1 \\ 1 & 1\end{bmatrix}[/tex]
    and
    [tex]\begin{bmatrix}3 & 2\\ 1 & 1\end{bmatrix}[/tex]
    must not have the same determinant. Is that true?

    I think you should to reread that problem.
     
  6. Jan 19, 2009 #5

    Hurkyl

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    No it doesn't. The question is asking you whether or not it is.
     
  7. Jan 19, 2009 #6
    Oh duh... That was stupid... It doesn't have to be. Thanks
     
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