Hello, I would like to have the answer to this exercise if possible?From question b to c if it is possible, because for a I was able to answer.I suspect that this exercise seems long. That's why I don't ask for the answers to the other questions.

It's bilinear algebra, master level.

Exercise 2.4. Let M be an orthogonal matrix in M2 (IR), meaning that it satisfies * MM = 12. 1/ First, suppose det M = 1. a/ Show that there exist two real numbers a and y satisfying a + y' = 1, such that My answer : det M = x2 -(-y?) = 1 , so det M = x3+yz = 1 M = b/ Show that there exists a real number 0 such that x = cos 0 and y = sin 60, then determine all solutions of the system cost(theta) cos 0 = x We know that cost(theta)+ sin?(theta) = 1 sin 0 = y in terms of 60. What are the eigenvalues of M? c/ Let SO2 (R) be the set of 2 x 2 orthogonal matrices with determinant 1. Show that the map from (R, +) to (SO2(IR), .), which maps 0 to the matrix cos 0 - sin 0\\ R(0) = sin 0 cos 0 is a group homomorphism, whose kernel is (2TZ, +). 2/ Now suppose det M = -1. Show that there exists a real number 0 such that cos 0 sin 0 M = sin 0 - cos 0 What are the eigenvalues of M? 3/ Show that any vector isometry in a 2-dimensional Euclidean space is either a rotation or a reflection