Question: 4. Given an n x n matrix A and scalar 1, define a subset of R by S= {VER | Av = Av}. Show that

4. Given an n x n matrix A and scalar 1, define a subset of R" by S= {VER" | Av = Av}. Show that S is a subspace.5. Let A and B be similar matrices. For any scalar A E R show that (a) A - XI and B - XI are similar, and (b) det(A - XI) = det(B - XI). Why would a mathematician find result (b) interesting?6. The linear operator of R2 that rotates every vector by an angle 0 in a counterclockwise direction is represented by the matrix A _ cos 6 sin 0 _ sin 3 cos 0 ' Find the eigenvalues and eigenvectors associated with A and interpret their meaning geometrically

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