Let be thegeneral linear group, that is, the group of all invertibles matrices over the finite field .

For the matrix over to be invertible. For it to be invertible it is necessary and sufficient that the first zero is non-zero and the second row is not a multiple of the first. The number of different possibilities for the first row is . While the number of multiples for the first row is therefore the number of rows that are not multiples is . Thus in total there are number of invertible matrices.

Now consider the group homomorphism defined through the determinant.

Notice that .

Therefore, by fundamental homomorphism theorem,

.

Thus, .

Note, that , thus,