Let p be a prime number. Let G be the group of all $latex 2 times 2 $ matrices over $latex Z_p $ 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 $latex GL_2 (F_p) $.
Solution:
First we find the order of General Linear Group $latex GL_2 (F_p) $ ; it is the group of all $latex 2 times 2 $ matrices with elements from a prime field (matrix multiplication defined accordingly) which are invertible.
$latex F_p $ has p elements (0, 1, 2, ... , p-1) hence to build a $latex 2 times 2 $ matrix there are p choices for each of the 4 spots creating $latex p^4 $ matrices in total. Since invertible matrices have non zero determinant, we deduct the number of matrices with zero determinant from $latex p^4 $ to get the order of $latex GL_2 (F_p) $.
Suppose a matrix with zero determinant is represented by $latex
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
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In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.