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.
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.
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.