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

• Apr 9th 2013, 04:07 AM
vick92414
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.
• Apr 9th 2013, 06:31 PM
chiro
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.
• Apr 9th 2013, 07:21 PM
johng
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.