How many invertible matrices are in $M_{2}(\mathbb{Z}_{11})$? I tried to solve this question but without a success.
How many invertible matrices are in $M_{2}(\mathbb{Z}_{11})$?
Thanks

The answer is $ (11^2 - 1)(11^2 - 11) $
The first term is how many non-zero vectors there are in $ (Z_{11})^2 $, the other one is how many there are that are linearly independent of the first one chosen.
You can easily generalize that fact: the power of all invertible $ n\times n$ matrices over a finite field $\mathbb{F}_p$ is
$$(p^n - 1)(p^n - p) \dots (p^n - p^{n-1})$$
Each of these tells us how many vectors there are that are linearly independent with the previous ones
