Homework SourseConsider the general linear group GLn R A Rnn detA is not e
Consider the general linear group GL(n, R) = {A Rn×n: detA is not equals 0} with the matrix multiplication as the group operation.
Prove that the subset SL(n, R) = {A Rn×n: detA = 1} is a normal subgroup of GL(n, R).
(I need the answer in details please!)
Solution
First observe that all matrices with det A = 1 are Orthogonal.
Firstly, te Identity matrix is in SL. Secondly, for any orthogonal matrix, det (A) = det (AT) => every element of SL inherits it\'s inverse from G.
Finally, det A = det B = 1. For orth matrics, det A. det B = det (AB) = 1. Thus SL is closed under multiplication.
Hence proving that SL is a subgroup of GL.
