let be the group of invertible matrices with entries in define the homomorphism by then and thus

to find look at the elements of as invertible transformations from to where is a vector space of dimension over let be a basis for let

then may be defined to be any non-zero element of so has options. now can be anything in so there are options for

hence and therefore

the above method can be used to, in general, find the order of the group of all matrices with determinant 1 and with entries from some finite field. (just copy the above proof!)