I need to find the relationship between the number of 2 x 2 matrices and the number of invertible 2 x 2 matrices over Zp where p is a prime number.
To help me, I took Z2 because 2 is a prime number. There appear to be 2^4=16 different matrices over Z2. Is this right?
Of these, I found 6 of them to be invertible.
Therefore, I think that the number of matrices is p^4. But I don't know the relationship between p and the number of invertible matrices. What is this?