a) Show that the set X of all 2x2 matrices with determinant = 1 is a smooth manifold. What is its dimension?
so i associated a 2x2 matrix with entries a11= x, a12 = y, a21 = w, and a22 = z to a vector in R^4 (x, y, w, z). so the function f (x, y, w, z) = xz-yw = 1. i took the derivative of this function and i got the row matrix [z -w -y x] and the linear transformation represented by this matrix is onto since at least one of the entries has to be nonzero for the determinant to equal 1 so this is a smooth manifold of dimension 3.
b) find a matrix in X that is closest to the matrix with entries a11 = 0, a12 = 1, a21 = 1, and a22 = 0.
the function i want to minimize is F(x,y,w,z) = x^2 + (y-1)^2 + (w-1)^2 + z^2. basically i want to minimize the distance between some matrix and the matrix described in part b).
so using lagrange multipliers i find the following equations:
2z=λxxz-yw = 1
maybe its just that its getting late but i seem to have trouble solving this system.
so i substituted from the first equation into the 4th one so i get 4z = (λ^2)z then i get (λ^2 - 4)z = 0. so if z = 0, then x = 0 as well and yw = -1. then i eliminate the λ between the 2nd and 3rd equations and substitute w = -1/y and after much simplifying i get y^4 - y^3 - y - 1 = 0 which i don't think has real solutions. going down the other route and assuming that λ = 2, then x = z and y + w = 1. i think i'm getting stuck around here as i don't know how to get numerical solutions. all i have is just variables in terms of other variables so far. is this the way to solve it or is there a better way?