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Math Help - skew-symmetric metrix

  1. #1
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    skew-symmetric metrix

    Hi i need some help solving the following problem:

    skew-symmetric metrix-untitled.png

    What i know:

    The following does not hold for p=2 since an inverse does not hold.

    but find it diffecult to show that this holds for primes p> 3.

    Thanks
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  2. #2
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    Quote Originally Posted by 1234567 View Post
    Hi i need some help solving the following problem:

    Click image for larger version. 

Name:	Untitled.png 
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ID:	19424

    What i know:

    The following does not hold for p=2 since an inverse does not hold.

    but find it diffecult to show that this holds for primes p> 3.

    Thanks


    For a prime p\geq 3 we have that 1\neq -1 and thus it's easy to show that dim S_n+dim A_n=dim M_n\,,\,\,M_n= all the square nxn matrices over the field...complete the proof now.

    Tonio
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  3. #3
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    how would one use dim S_n+dim A_n=dim M_n\,,\,\,M_n= all the square nxn matrices over the field,
    I thought you hav to show

    1.  S_n+A_n= M_n\,,
    2. S_n n A_n= {0}\,,
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  4. #4
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    Quote Originally Posted by 1234567 View Post
    how would one use dim S_n+dim A_n=dim M_n\,,\,\,M_n= all the square nxn matrices over the field,
    I thought you hav to show

    1.  S_n+A_n= M_n\,,
    2. S_n n A_n= {0}\,,

    The second condition is trivial, so \dim S_n+\dim A_n=\dim(S_n+A_n)-\dim(S_n\cap A_n)=\dim(S_n+A_n) , and thus

    proving what I told you we get S_n+A_n=M_n

    Tonio
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