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.

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.

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 (p^{2}-1)(p^{2}-p). You can look on the web for "general linear group"; in particular, the article on wikapedia answers your question and more.