Let p be a prime number. Let G be the group of all matrices over with determinant 1 under matrix multiplication. Then the order of G is
(Additional Problem - this did not come in IIT JAM 2013): Find the order of general linear group .
First we find the order of General Linear Group ; it is the group of all matrices with elements from a prime field (matrix multiplication defined accordingly) which are invertible.
has p elements (0, 1, 2, ... , p-1) hence to build a matrix there are p choices for each of the 4 spots creating matrices in total. Since invertible matrices have non zero determinant, we deduct the number of matrices with zero determinant from to get the order of .
Suppose a matrix with zero determinant is represented by $latex