Algebra and Geometry | Part IA, 2005

Let AA be a 3×33 \times 3 real matrix such that det(A)=1,AI\operatorname{det}(A)=-1, A \neq-I, and ATA=IA^{T} A=I, where ATA^{T} is the transpose of AA and II is the identity.

Show that the set EE of vectors xx for which Ax=xA x=-x forms a 1-dimensional subspace.

Consider the plane Π\Pi through the origin which is orthogonal to EE. Show that AA maps Π\Pi to itself and induces a rotation of Π\Pi by angle θ\theta, where cosθ=12(trace(A)+1)\cos \theta=\frac{1}{2}(\operatorname{trace}(A)+1). Show that AA is a reflection in Π\Pi if and only if AA has trace 1 . [You may use the fact that trace(BAB1)=trace(A)\operatorname{trace}\left(B A B^{-1}\right)=\operatorname{trace}(A) for any invertible matrix B.]

Prove that det(AI)=4(cosθ1)\operatorname{det}(A-I)=4(\cos \theta-1).

Typos? Please submit corrections to this page on GitHub.