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Math Help - Matrix Groups

  1. #1
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    Matrix Groups

    Any help with the following problem is greatly appreciated:



    For the part (a) of this problem, I must show that M_{a,p} belongs to the general linear group of 2x2 matrices over p (a prime) iff a \bmod\ p \neq p-1.

    I know from a definition that GL(2,F)= \{ A\in Mat_{2} (F) | det(A) \neq 0 \in F \}.

    So here we must have det \begin{bmatrix}1 & (a \bmod\ p)\\p-1 & 1\end{bmatrix} = (a \bmod\ p) (p-1) \neq 0

    I'm kind of stuck here, what can I do next?
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  2. #2
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    First, you have calculated the derivative incorrectly- you forgot the main diagonal.
    \left|\begin{array}{cc}1 & a (mod p) \\ p-1 & 1\end{array}\right|= 1- (p-1)[a (mod p)].

    That will be 0 if and only if (p-1)[a (mod p)]= 1.
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  3. #3
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    Thanks, that was a typo. So, we must have that 1- (p-1)(a \bmod\ p) \neq 0 \iff (p-1)(a \bmod\ p) \neq 1. So I guess (p-1) must be equal to -1 in \mathbb{Z}_p. Therefore p-1 = -1 \neq a \bmod\ p.

    But how do I prove that (p-1)=-1?


    And for part (b) I think they are asking for the inverse of

    M_{10,p} = \left|\begin{array}{cc}1 &  (10 \bmod\ p) \\ p-1 & 1\end{array}\right|

    Then the inverse would be of the form:

    \frac{1}{det(M_{10,p})} \left|\begin{array}{cc}1 &  -(10 \bmod\ p) \\ 1-p & 1\end{array}\right|

    But what values do I use for det(M_{10,p}) and 10 mod p?
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  4. #4
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    Quote Originally Posted by demode View Post
    Thanks, that was a typo. So, we must have that 1- (p-1)(a \bmod\ p) \neq 0 \iff (p-1)(a \bmod\ p) \neq 1. So I guess (p-1) must be equal to -1 in \mathbb{Z}_p. Therefore p-1 = -1 \neq a \bmod\ p.

    But how do I prove that (p-1)=-1?


    !!!... As p=0\!\!\pmod p , obviously p-1=0-1=-1\!\!\pmod p ...!

    Tonio



    And for part (b) I think they are asking for the inverse of

    M_{10,p} = \left|\begin{array}{cc}1 &  (10 \bmod\ p) \\ p-1 & 1\end{array}\right|

    Then the inverse would be of the form:

    \frac{1}{det(M_{10,p})} \left|\begin{array}{cc}1 &  -(10 \bmod\ p) \\ 1-p & 1\end{array}\right|

    But what values do I use for det(M_{10,p}) and 10 mod p?
    .
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  5. #5
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    Thank you. And for part (b) I know the inverse would be of the form

    \frac{1}{1-(p-1)(10 \bmod\ p)} \begin{bmatrix} 1&-(a \bmod\ p)\\ -(p-1)& 1 \end{bmatrix}

    Right? But what is 10 \bmod\ p when p \in \{ 3,7 \}? 7 or 3? Also I need to know (p-1). (I must know the value of the determinant in order to find its inverse).
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