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Math Help - the number of invertible elements in the ring of all 2*2 matrices over Zp

  1. #1
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    the number of invertible elements in the ring of all 2*2 matrices over Zp

    Who can help me solution this question?
    The number of invertible elements in the ring of all 2*2 matrices over Zp.
    P is a prime number.
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  2. #2
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    Re: the number of invertible elements in the ring of all 2*2 matrices over Zp

    Hey vick92414.

    The determinant will be ad - bc != 0. This implies ad != bc.

    This implies that for a factorization you have ad is co-prime to bc.

    You could use the prime distribution theorem (i.e. the one that Gauss conjectured) to quantify this probability as an approximation.
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  3. #3
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    Re: the number of invertible elements in the ring of all 2*2 matrices over Zp

    Hi,
    Rather than answer your question in detail, let me just say the answer is (p2-1)(p2-p). You can look on the web for "general linear group"; in particular, the article on wikapedia answers your question and more.
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